Extension from Precoloured Sets of Edges

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.


Introduction
Conjecture 3 strengthens Proposition 2 in two ways: we impose a weaker constraint on the distance between precoloured edges, and we use a smaller palette. Evidently, we believe that in edge-precolouring the distance requirement ought to be not as strong as it is for vertex-precolouring extension. In Section 2, however, we show how Conjecture 3 becomes false if we are allowed to precolour a distance-1 rather than a distance-2 matching. Note that Conjecture 3 easily becomes false, even for trees, if we replace the palette K by [∆(G)] or by [χ (G)], where χ (G) is the chromatic index of G. For instance, consider stars with each edge subdivided exactly once; see Figure 1.
In another direction, one might wonder if a strong enough distance requirement on the precoloured matching permits us to take a smaller palette, like [∆(G)] or [χ (G)]. This fails however, even for bipartite graphs, as we now show.
First, for any positive integer m, let D m denote the bipartite graph on vertex set {x} ∪ A x ∪ B ∪ A y ∪ {y}, where |A x | = |A y | = m and |B| = 2m − 1, and whose edge set is the set of all pairs between {x} ∪ B and A x and between {y} ∪ B and A y . Let us observe an easy property of the graph D m : in any proper edge-colouring of D m with colours from [2m], there must be at least one edge of colour 1 incident to x or y. For otherwise, since each vertex in A x has degree 2m, there must be m edges of colour 1 between A x and B; similarly, there must be m edges of colour 1 between A y and B. But this implies that there are 2m distinct edges of colour 1 incident to the 2m − 1 vertices in B, which means that a vertex of B is incident to two edges of colour 1, a contradiction.
Next, for any positive integers , m, let G m, be the graph formed by taking disjoint copies H 1 , . . . , H of D m with vertex sets labelled {x i } ∪ A x i ∪ B i ∪ A y i ∪ {y i }, identifying y i with x i+1 for all i = 1, . . . , − 1, and then adding two new vertices x and y and two new edges x x 1 and y y . See Figure 2 for a depiction of G 3,2 . It is straightforward to check that G m, is bipartite, has maximum degree 2m, and that the edges x x 1 and y y are at distance 4 + 1 in G m, . Consider a precolouring of G m, from the palette [2m] = [∆(G m, )] = [χ (G m, )] in which the edges x x 1 and y y are precoloured 1. Suppose, for a contradiction, that there is a proper extension of this precolouring. Then there can be no edge of colour 1 between A y 1 and B 1 . By our observation about D m , there must be an edge of colour 1 between A y 1 and y 1 = x 2 . It follows by an induction (via copies of D m ) that there is an edge of colour 1 between A y and y . Since y y is precoloured 1, we have arrived at our desired contradiction.
If true, Conjecture 3 would extend Vizing's theorem [37], which is independently due to Gupta, cf. [18]. A variant of Conjecture 3 was proved by Berge and Fournier [7,Cor. 2] -they showed that extension is guaranteed, even from precoloured distance-1 matchings, provided that all edges of the matching have been precoloured with the same colour.
In this paper, we prove several special cases of Conjecture 3, in particular, for bipartite multigraphs, subcubic multigraphs, and planar graphs of large enough maximum degree. Indeed, for these classes we show that Conjecture 3 holds even when the precoloured set is allowed to be a distance-1 matching. Moreover, we prove a variant of Conjecture 3, where the extended edge-colouring avoids some prescribed colours on a (distance-1) matching. We discuss this further in Subsection 1.1. However, first allow us to place the conjecture in context by giving some preliminary observations. By the following easy observation, Conjecture 3 is also related to list edge-colouring, and therefore to the List Colouring Conjecture (LCC), which states that ch (G) = χ (G) for any multigraph G (where ch (G) is, as usual, the list chromatic index of G).
For a non-precoloured edge, we define its precoloured degree as the number of adjacent precoloured edges.
Observation 4. Let G be a multigraph with list chromatic index ch (G). For a positive integer k, take the palette as K = [ch (G) + k]. If G is properly precoloured so that the precoloured degree of any non-precoloured edge is at most k, then the precolouring can be extended to a proper edge-colouring of all of G.
So, if we assume that the LCC holds, then the following weak form of Conjecture 3 holds as well: using the palette K = [∆(G) + µ(G) + 1], any precoloured distance-2 matching extends to all of G. Observation 4 follows from a more refined statement we will give in Section 3.
Due to the remarkable work of Kahn [22,23,24] on edge-colourings and list edgecolourings of (multi)graphs, not only does an asymptotic form of Conjecture 3 hold, but so does a precolouring extension of an asymptotic form of the Goldberg-Seymour Conjecture (which we review in Subsection 1.2). Kahn's theorem and Observation 4 together imply the following.
Proposition 5. For any ε > 0, there exists a constant C ε such that the following holds. For any multigraph G with χ (G) C ε , any precoloured matching using the palette K = [(1 + ε)χ (G)] can be extended to a proper edge-colouring of all of G. If we replace χ (G) in the statement above by ∆(G) + µ(G) or by the Goldberg-Seymour bound, then the statement remains valid, either due to Vizing's theorem or due to another theorem of Kahn.
One of our motivations for the formulation and study of Conjecture 3 comes from the close connections with vertex-precolouring and with the LCC.

Main Results
Although it appears that the LCC and our conjecture are independent statements, we have obtained several results corresponding to specific areas of success in list edge-colouring. In summary, we confirm Conjecture 3 for bipartite multigraphs, subcubic multigraphs, and planar graphs of large enough maximum degree. We also obtain a precolouring extension variant of Shannon's theorem, and we confirm a relaxed version of Conjecture 3, where the extended edge-colouring avoids some prescribed colours on a matching. Furthermore, all of these partial results hold in the more general context where the precoloured set is allowed to be a distance-1 matching, rather than the distance-2 matching required by Conjecture 3. In fact, in this section we mostly present our main results restricted to precoloured matchings, to aid clarity, even when yet more general statements hold.
Our first result is an edge-precolouring extension of Kőnig's theorem that any bipartite multigraph G is ∆(G)-edge-colourable, whereas the subsequent result is an edgeprecolouring analogue of Shannon's theorem that any multigraph G is 3 2 ∆(G) -edgecolourable.
Theorem 6. Let G be a bipartite multigraph with maximum degree ∆(G). With the palette K = [∆(G) + 1], any precoloured matching can be extended to a proper edge-colouring of all of G. Figures 1 and 2, the palette size in Theorem 6 is sharp.

As indicated in
Theorem 7. Let G be a multigraph with maximum degree ∆(G). With the palette K = 3 2 ∆(G) + 1 2 , any precoloured matching can be extended to a proper edge-colouring of all of G.
Due to the Shannon multigraphs, this last statement is sharp if ∆(G) is even, and within 1 of being sharp if ∆(G) is odd. Theorems 6 and 7 are proved in Section 3 using powerful list colouring tools developed by Borodin, Kostochka and Woodall [11].
The following theorem concerns multigraphs that are subcubic, i.e., of maximum degree at most 3. Note that Theorem 8 improves upon Theorem 7 for ∆(G) = 3.
Theorem 8. Let G be a subcubic multigraph. With the palette K = [4], any precoloured matching can be extended to a proper edge-colouring of all of G.
The example we give in Section 2 shows that 3 is the largest value of ∆(G) for which we are guaranteed that the palette [∆(G) + 1] is enough to extend every precoloured matching to a proper edge-colouring of the whole graph. In other words, Theorem 8 is the electronic journal of combinatorics 25(3) (2018), #P3.1 best possible with respect to ∆(G). A form of Theorem 8, for subcubic simple graphs and with a distance condition on the precoloured matching, was observed by Albertson and Moore [3]. Although the LCC remains open for subcubic graphs, Juvan, Mohar anď Skrekovski [20] have made a significant attempt. They showed that for any subcubic graph G, if lists of 3 colours are given to the edges of a subgraph H with ∆(H) 2 and lists of 4 colours to the other edges, then G has a proper edge-colouring using colours from those lists.
Theorem 8 is a direct corollary of the following theorem, which may be of interest in its own right. Its proof uses a degree-choosability condition and can be found in Section 4.
Theorem 9. Let G be a connected multigraph with maximum degree ∆(G). Choose a nonnegative integer k such that ∆(L(G)) ∆(G) + k, and take the palette as K = [∆(G) + k]. If G is properly precoloured so that the precoloured degree of any vertex is at most k, then the precolouring can be extended to a proper edge-colouring of all of G, except in the following cases: (a) k = 0 and G is a simple odd cycle; (b) G is a triangle with edges of multiplicity m 1 , m 2 , m 3 and k = min{m 1 , m 2 , m 3 } − 1.
Note that, when restricted to precoloured matchings, this theorem produces weak or limited bounds for larger maximum degree. On the other hand, if we replace every precoloured edge in the example of Figure 1 by a precoloured multi-edge of multiplicity k (or k + 1) and a precolouring from [k] (or [k + 1]), we see that the palette bound (or precoloured degree condition) is best possible.
In the case where k = ∆(L(G)) − ∆(G) in Theorem 9, the number of colours used is equal to the maximum degree of the line graph. In that sense the theorem can be considered as a precolouring extension of Brooks's theorem restricted to line graphs. It is relevant to mention that vertex-precolouring extension versions of Brooks's theorem [2,6,28] require, among other conditions, a large minimum distance between the precoloured vertices.
The class of planar graphs could be of particular interest. There is a prominent line of work on (list) edge-colouring for this class, which we discuss further in Subsection 1.2 and Section 5. Our main contributions to this area are the following results, the second one of which can be viewed as a strengthening of another old result of Vizing [38], provided the graph's maximum degree is large enough.    Table 1: Summary of edge-precolouring extension results for planar graphs with maximum degree ∆, when a distance-t matching M is precoloured using the palette K. See Section 5 for further details how these results can be obtained.
Due to the trees exhibited in Figure 1, the palette size in Theorem 10 cannot be reduced, while the minimum distance condition in Theorem 11 cannot be weakened. In Section 5, we give some more results on when it is possible for a precoloured matching in a planar graph to be extended. A summary of the results is given in Table 1.
Suppose that we would go to any means to obtain an extension form of Vizing's theorem, say, by weakening the precolouring condition. We still let K = [K] be a palette of available colours. Given a subset S ⊆ E of edges and an arbitrary (i.e., not necessarily proper) colouring of elements of S using only colours from K, is there a proper colouring of all edges of G (using colours from K) that differs from the given colouring on every edge of S? We may consider the coloured set S as a set of forbidden (coloured) edges, while the full colouring, if it can be produced, is called an avoidance of the forbidden edges. We can show the following result, which, while it is in one sense weaker than the statement in Conjecture 3, is directly implied by neither the LCC nor other existing precolouring results, implies Vizing's theorem, and provides further evidence in support of Conjecture 3. (This result was stated as a conjecture in an earlier version of this paper.) Theorem 12. Let G be a multigraph with maximum degree ∆(G) and maximum multiplicity µ(G). Using the palette K = [∆(G)+µ(G)], any forbidden matching can be avoided by a proper edge-colouring of all of G.
We use an aforementioned result of Berge and Fournier and a recolouring argument to prove this theorem in Section 6.
Some basic knowledge of edge-colouring is a prerequisite to the consideration of edgeprecolouring extension problems -we provide some related background in the next subsection. To our frustration, many of the major methods for colouring edges (such as Kempe chains, Vizing fans, Kierstead paths, Tashkinov trees) seem to be rendered useless by precoloured edges. Though Conjecture 3 may at first seem as if it should be an "easy extension" of Vizing's theorem, it might well be difficult to confirm (if true). We are the electronic journal of combinatorics 25(3) (2018), #P3.1 keen to learn of related edge-precolouring results independent of current list colouring methodology.

Further Background
Edge-colouring is a classic area of graph theory. We give a quick overview of some of the most relevant history for our study. The reader is referred to the recent book by Stiebitz, Scheide, Toft and Favrholdt [34] for detailed references and fuller insights. The lower bound χ (G) ∆(G) is obviously true for any multigraph G. Close to a century ago, Kőnig proved that all bipartite multigraphs meet this lower bound with equality. Shannon [32] in 1949 proved that χ (G) 3 2 ∆(G) for any multigraph G. Somewhat later, Gupta (as mentioned in [18]) and, independently, Vizing [37] proved that χ (G) ∆(G) + µ(G) for any multigraph G, so χ (G) ∈ {∆(G), ∆(G) + 1} if G is simple. Both the Shannon bound and the Gupta-Vizing bound are tight in general due to the Shannon multigraphs, which are triangles whose multi-edges have balanced multiplicities. (Note however that the latter bound can be improved for specific choices of ∆(G) and µ(G), as described in the work of Scheide and Stiebitz [30].) A notable conjecture on edge-colouring arose in the 1970s, on both sides of the iron curtain. The Goldberg-Seymour Conjecture, due independently to Goldberg [17] and Seymour [31], asserts that χ (G) ∈ { ∆(G), ∆(G) + 1, ρ(G) } for any multigraph G, where The parameter ρ(G) is a lower bound on χ (G) based on the maximum ratio between the number of edges in H and the number of edges in a maximum matching of H, taken over induced subgraphs H of G. This conjecture remains open and is regarded as one of the most important problems in chromatic graph theory. Perhaps the most outstanding progress on this problem is due to Kahn [23], who established an asymptotic form.
The list variant of edge-colouring can be traced as far back as list colouring itself. The concept of list colouring was devised independently by Vizing [39] and Erdős, Rubin and Taylor [14], with the iron curtain playing its customary role here too. The List Colouring Conjecture (LCC) was already formulated by Vizing as early as 1975 and was independently reformulated several times, a brief historical account of which is given by, e.g., Häggkvist and Janssen [19]. For more on the LCC, particularly with respect to the probabilistic method, consult the monograph of Molloy and Reed [27]. The results on the LCC most relevant to our investigations also happen to be two of the most striking, both from the mid-1990s. First, Galvin [16] used a beautiful short argument to prove Dinitz's Conjecture (concerning the extension of arrays to partial Latin squares), which at the same time confirmed the LCC for bipartite multigraphs. Not long after Galvin's work, Kahn applied powerful probabilistic methods, with inspiration from extremal combinatorics and statistical physics, to asymptotically affirm the LCC [23,24]. For more background on Kahn's proof, related methods, and improvements, consult [19,26,27].
Inspiration for this class of problems may also be taken from list vertex-colouring. For instance, we utilise a degree-choosability criterion due independently to Borodin [9] and the electronic journal of combinatorics 25(3) (2018), #P3. 1 Erdős, Rubin and Taylor [14]. See for example a survey of Alon [4] for an excellent (if older) survey on list colouring in somewhat more generality. We should mention that part of the motivation for studying list colouring was to use it to attack other, less constrained colouring problems. The connection has gone back in the other direction as well, as precolouring extension demonstrates.
Activity in the area of precolouring extension increased dramatically as a result of the startling proof by Thomassen of planar 5-choosability [35]; a key ingredient in that proof was a particular type of precolouring extension from some pair of adjacent vertices, according to a specific planar embedding. A little bit later, Thomassen asked about precolouring extension for planar graphs under a more general setup [36]. Eliding the planarity condition, Albertson [1] quickly answered Thomassen's question and proved more: in any k-colourable graph, for any set of vertices with pairwise minimum distance at least 4, any precolouring of that set from the palette [k +1] can be extended to a proper colouring of the entire graph. (This implies Proposition 2 above.) Since Albertson's seminal work, a large body of research has developed around precolouring extension. But this research has focused almost exclusively on extension of vertex-colourings. One of the few papers we are aware of that deals with edge-precolouring extension is by Marcotte and Seymour [25], in which a different type of necessary condition for extension is studied -curiously, this paper predates the above mentioned activity in vertex-precolouring.
For planar graphs, there has been significant interest in both edge-colouring and list edge-colouring. It is known that planar graphs G with ∆(G) 7 satisfy χ (G) = ∆(G). This was proved in 1965 by Vizing [38] in the case ∆(G) 8, and much later by Sanders and Zhao [29] for ∆(G) = 7. We remark that Theorem 11 strengthens this for ∆(G) somewhat larger. Vizing conjectured that the same can be said for planar graphs G with ∆(G) = 6, but this long-standing question remains open. Vizing also noted that not every planar graph G with ∆(G) ∈ {4, 5} is ∆(G)-edge-colourable. Regarding list edge-colouring, Borodin, Kostochka and Woodall [11] proved the LCC for planar graphs with maximum degree at least 12, i.e., they proved that such graphs have list chromatic index equal to their maximum degree. The LCC remains open for planar graphs with smaller maximum degree, though it is known that if ∆(G) 4 or ∆(G) 8, then ch (G) ∆(G) + 1 (Juvan, Mohar andŠkrekovski [21] for ∆(G) 4; Bonamy [8] for ∆(G) = 8; Borodin [10] for ∆(G) 9). As noted above, it is not true that planar graphs G with ∆(G) ∈ {4, 5} are always ∆(G)-edge-choosable.

Necessity of the Distance-Condition
In this section, we show that if we omit the distance-2 condition on the precoloured matching then Conjecture 3 becomes false whenever ∆(G) 4. For each t 3, we construct a graph G t of maximum degree t + 1 with the property that, using the palette K = [t + 2], there is a matching M and a precolouring of M that cannot be extended to a proper edge-colouring of all of G t .
let H t be the graph obtained from K t,t by subdividing one edge.
Proof. Since H t has 2t + 1 vertices, its largest matching has size t. Since H t has t 2 + 1 edges, we cannot cover all the edges with t matchings.
Let A, B ⊆ V (H t ) be the original partite sets of K t,t , so that A and B are independent sets of size t in H t , and the only vertex of H t not contained in A ∪ B is the vertex of degree 2.
Let H t be the graph obtained from H t by attaching a pendant edge to each vertex of H t , and for each v ∈ V (H t ), let v be the other endpoint of the pendant edge at v. Finally, We precolour the matching M 0 by colouring vv colour 1 if v ∈ A, and colouring vv colour 2 otherwise. Now we define the full graph G t by taking t + 1 disjoint copies of H t , and adding a new vertex v * adjacent to the unique vertex of degree 3 in each copy of H t . The precoloured matching M in G t is just the union of each precoloured matching M 0 in each copy of H t , with the same precolouring. Figure 3 shows G 3 .
Theorem 14. For every t 3, using the palette , the precolouring of the matching M as described above cannot be extended to a proper edgecolouring of all of G t .
Proof. Suppose to the contrary that G t has an edge-colouring from K that extends the precolouring of M . Since every neighbour of v * has an incident edge precoloured 2, no edge incident to v * can be coloured 2. Therefore, since d(v * ) = t + 1, each of the t + 1 colours excluding 2 is used exactly once on the edges incident to v * . In particular, some edge e incident to v * has colour 1. Let H be the copy of H t containing the other endpoint of e. Observe that no edge of H can be coloured 1 or 2: every edge joining A and B has an edge precoloured 1 at one endpoint and an edge precoloured 2 at the other, while the electronic journal of combinatorics 25(3) (2018), #P3.1 the vertex of degree 2 in H is incident to an edge precoloured 2 as well as the edge e coloured 1. Hence all edges of H use only the t remaining colours. Since χ (H t ) = t + 1 by Lemma 13, this is impossible.

Extensions of Kőnig's and Shannon's Theorems
Theorem 15 below implies Theorem 6, and hence verifies Conjecture 3 for bipartite multigraphs. Theorem 16 implies Theorem 7. Recall that the precoloured degree of a vertex is the number of incident precoloured edges.
Theorem 15. Let G be a bipartite multigraph and k 1. Take the palette as K = [∆(G) + k]. If G is properly precoloured so that the precoloured degree of any vertex is at most k, then this precolouring can be extended to a proper edge-colouring of all of G.
Theorem 16. Let G be a multigraph and k 1. Take the palette as K = 3 2 ∆(G)+ 1 2 k . If G is properly precoloured so that the precoloured degree of any vertex is at most k, then this precolouring can be extended to a proper edge-colouring of all of G.
The two results are corollary to two theorems of Borodin, Kostochka and Woodall [11].
if for any assignment of lists in which every edge e receives a list of size at least f (e), there is a proper edge-colouring of G using colours from the lists.
Theorem 18 (Borodin, Kostochka & Woodall [11]). Let G be a multigraph, and set Note that Theorem 17 is a strengthening of Galvin's theorem; while Theorem 18 is a list colouring version of Shannon's theorem (and in fact follows from Theorem 17).
In our proofs of Theorems 15 and 16, we use the following refinement of Observation 4. Given a graph G and an edge e, the degree d G (e) of e is the number of edges adjacent to e in G.
f (e), then the precolouring can be extended to a proper edge-colouring of all of G.
Proof of Theorems 15 and 16. Assume that G = (V, E) and S ⊆ E is the set of precoloured edges. Set E = E \ S, G = (V, E ) and G = (V, S). Consider any uncoloured edge e = uv ∈ E , and assume that d G (u) d G (v). In the bipartite case, since ∆ Theorem 15 follows by combining Observation 19 and Theorem 17.
In the general case (Theorem 16) we obtain This time combining Observation 19 with Theorem 18 completes the proof.

An Approach using Gallai Trees
In this section, we use a result due independently to Borodin [9] and to Erdős, Rubin and Taylor [14]. This is a list version of an older result of Gallai [15] on colour-critical graphs.
A connected graph all of whose blocks are either complete graphs or odd cycles is called a Gallai tree. With this we prove Theorem 9, which implies Theorem 8.
Proof of Theorem 9. Assume to the contrary that the connected multigraph G and the non-negative integer k satisfy ∆(L(G)) ∆(G) + k, but that, using the palette K = [∆(G) + k], there is a proper edge-precolouring of G of the required type that does not extend to a proper edge-colouring of G. For a vertex v, let K(v) ⊆ K be the set of colours appearing on the precoloured edges incident with v, and set k(v) = |K(v)|.
Let G be obtained from G by deleting all precoloured edges. To each edge e = uv in G , we assign a list (e) containing those colours in K not appearing on precoloured edges adjacent to e in G. For any edge e = uv in G we obtain, using that ∆ Since there is no extension of the precolouring of L(G) to a full colouring of L(G), it follows that L(G ) is not vertex-choosable with the lists (e), for e ∈ E(G ). In particular, there is a component C of G such that L(C ) is not vertex-choosable with the lists (e), for e ∈ E(C ). By Theorem 20, L(C ) must be a Gallai tree such that | (e)| = d L(C ) (e) for every e. This also means that we must have equality in all inequalities used to derive (1); in particular: for all e ∈ E(C ): So, analogously to (1) above, we infer that for each edge e = uv in C the order of (e) is at least the degree in C of each of its end-vertices: We require the following statements. We continue by considering the case that C is not an odd cycle. Since line graphs are claw-free, it follows that odd cycle blocks of length at least five are impossible in L(C ). We deduce that all blocks of L(C ) are cliques. The only way that a leaf block B of L(C ) could be part of a nontrivial block structure is if it corresponds to a set of edges in C that are all incident with a unique vertex, with one of the edges corresponding to the cut-vertex of B. This is ruled out by Claim 21. We conclude that L(C ) must itself be a clique. In turn, the only way that a line graph L(C ) of a multigraph is a clique is if C is a star or a triangle, with possibly multiple edges. The first option is ruled out by Claim 21, so C must be a triangle, possibly with multi-edges. that C is not edge-choosable is if all the lists are the same. This also means that the sets K(u) ∪ K(v), K(u) ∪ K(w) and K(v) ∪ K(w) are the same.
Let A(u) be the set of colours that appear on precoloured edges incident with u, but not with v or w; define A(v) and A(w) analogously. (In other words, these are colours on the edges that connect C to the rest of the graph G.) Let D be the set of colours that appear on precoloured edges with end-vertices contained in {u, v, w}. From (2a) and (2b) we deduce that | (e)| = d L(C ) (e) for every edge e in C , which, applied to an edge between u and v, implies that Now recall that all edges in C must have the same list. Consequently, the disjointness of the sets A(u), A(v) and A(w) implies that these three sets are empty. Thus we find that there are no precoloured edges between any of u, v, w and the rest of the graph. Since

Planar Graphs
In this section, for brevity we usually write ∆ for ∆(G).
In the next subsection we prove Conjecture 3 for planar graphs of large enough maximum degree (at least 17), which is the assertion of Theorem 10. As mentioned earlier, the LCC is known to hold for planar graphs with maximum degree at least 12. This is yet another result of Borodin, Kostochka and Woodall [11]: they indeed show that ch (G) ∆ for such graphs G. Combining this with Observation 4 gives the bounds in lines 3 and 7 of Table 1. Since the former bound will be useful for us later on, let us state it formally. Proposition 23. Let G be a planar graph with maximum degree ∆(G) 12. Using the palette K = [∆(G) + 2], any precoloured matching can be extended to a proper edgecolouring of all of G.
Borodin [10] showed that ch (G) ∆ + 1 for planar graphs G of maximum degree ∆ 9. Recently, Bonamy [8] extended this last statement to the case ∆ = 8. Combining this result with Observation 4 implies that for planar graphs with maximum degree ∆ 8 a precoloured matching can be extended to a proper colouring of the entire graph with the palette [∆ + 3], while a precoloured distance-2 matching can be extended with the palette [∆ + 2].
For smaller values of ∆, we can use Theorems 7 and 8, and the result of Juvan, Mohar andŠkrekovski [21] that ch (G) ∆(G) + 1 for a planar graph G with ∆(G) 4, to achieve several of the bounds in Table 1. In particular, it follows that ∆ + 4 colours suffice for any planar graph with maximum degree ∆.
The final proof we present is of Theorem 11. As discussed in Subsection 1.2, Vizing conjectured [38] that any planar graph with maximum degree ∆ 6 has a ∆-edge-  Figure 1 show that this statement is false if we allow an adversarial precolouring of a distance-2 matching. But does it remains true with the adversarial precolouring of any distance-3 matching? We prove that this is indeed the case if ∆ 23. We expect that this lower bound on ∆ can be reduced, though, as noted before, certainly not below 6.

colouring. The examples in
The proofs of Theorems 10 and 11 can be found in the next two subsections. They use a common framework, terminology and notation, which we outline now. Note that both adapt a nice trick of Cohen and Havet [12], which shortens the argument considerably.
Whenever considering a planar graph G, we fix a drawing of G in the plane. (So we really should talk about a plane graph.) Because of this fixed embedding we can talk about the faces of the graph. If G is connected, then the boundary of any face f forms a closed walk W f . We adopt the following notation to classify the vertices of a graph G according to their degree and their incidence with vertices of degree 1. Let V i be the set of vertices of degree i. Also, identify by T i ⊆ V i the set of those vertices of degree i that are adjacent to a vertex of degree 1, and set U i = V i \ T i . Write T = i 1 T i and U = V (G) \ T . We also adopt the shorthand notation V [i,j] , U [i,j] and T [i,j] to mean, respectively, the sets of vertices in V , U and T with degrees between i and j inclusively.

Proof of Theorem 10
If G is not connected, then we extend the edge-colouring one component at a time. The colouring of a component C with ∆(C) 16 can be extended using the results on lines 1 -3 of Table 1. Next, the statement of Theorem 10 is true for graphs with maximum degree 17 and exactly 17 edges. We use induction on E(G), and proceed with the induction step. So we may assume that G is connected and has at least 18 vertices, since ∆ 17. Let M be a precoloured matching.
We first observe that Indeed, suppose that the inequality does not hold for some edge uv / ∈ M . Then, by induction if ∆(G − uv) 17 and by Proposition 23 if ∆(G − uv) = 16, there exists an extension of M to a colouring of all G − uv using the palette K. Since at most ∆ colours are used on the edges adjacent to uv, we can easily extend the colouring further to uv. It follows from this observation that G has no vertices of degree 2, that every vertex with degree 1 is incident with an edge in M and that any vertex has at most one neighbour of degree 1. We will use these facts often without reference in the remainder of the proof.
For If v ∈ T has a (unique) neighbour in V 1 , then we always choose v 1 to be this neighbour. In that case we have f d(v) = f 1 ; we denote that face by f 1 again. Note that it is possible for other faces to be the same as well (if v is a cut-vertex), but we will not identify those multiple names of the same face. So, if v ∈ U , then the faces around v in consecutive order are f 1 , f 2 , . . . , f d(v) ; while, if v ∈ T , then the faces around v are f 1 , f 2 , . . . , f d(v)−1 .
Proof. Consider the set F of edges in E(G) \ M with one end-vertex in V 3 and the other in V ∆ . The subgraph with vertex set V 3 ∪ V ∆ and edge set F is bipartite; we assert it is acyclic. For suppose there exists an (even) cycle C with E(C) ⊆ F . By induction if ∆(G − E(C)) 17 and by Proposition 23 if ∆(G − E(C)) ∈ {15, 16}, we can extend the precolouring of M to G − E(C) using the palette K. But then we can further extend this colouring to the edges in C, since each edge in C is adjacent to only ∆ − 1 coloured edges, and even cycles are 2-edge-choosable.
Since each vertex in V 3 is incident with at least two edges in F , we have |V ∆ | + |V 3 | > |F | 2|V 3 |. The claim follows.
We use a discharging argument to continue the proof of the theorem. First, let us assign to each vertex v a charge α1: α(v) = 3d(v) − 6, and to each face f a charge α2: α(f ) = −6.
For each vertex v we define β(v) as follows.
It follows from Claim 24 that v β(v) < 0. Finally, from Euler's formula for simple plane graphs, we obtain Thus, in order to reach a contradiction, it is enough to show that for every vertex v: and that for every face f : Let f be a face. As G is simple, |V − (f )| 3. Since α(f ) = −6, to establish (6) it is enough to show that δ(f ) 6. Let v be a vertex in V − (f ) for which δ v (f ) is minimum. If δ v (f ) · |V − (f )| 6, then (6) clearly holds, and so we only need to deal with cases δ1 -δ4. Also, if v ∈ T [7,∆−2] ∪ U [6,∆−2] , then δ3 and δ4 give δ v (f ) 2, and hence again (6) is verified.
If v ∈ T [3,4] , then δ v (f ) = 1 by δ1 or δ3 and, by (4), the neighbours u and w of v in V − (f ) have degree at least ∆ − 1. If |V − (f )| = 3, then δ6 applies to both u and w, so If |V − (f )| 4, then δ9 applies to both u and w, so δ u (f ) = δ w (f ) = 2, while a fourth vertex z in V − (f ) satisfies δ z (f ) 1 by the definition of v. So (6) always follows.
Recall that G has no vertices of degree 2. If d(v) = 3, then α(v) = 3, while β(v) = 2 by β2. If v ∈ T 3 , then γ(v) = −3 and δ1 implies that δ(v) = −2. If v ∈ U 3 , then γ(v) = 0 and δ2 implies that δ(v) = −5. This confirms (5) Similarly, if v ∈ U , then γ(v) = 0, and δ4 implies that . This proves (5) for those vertices v. Now suppose that d(v) ∆ − 1. As a next step towards proving (5), we consider the average value of δ f (v) over the faces incident with v. For convenience, set Proof. To obtain the desired bound, we group some of the faces around v into disjoint consecutive triples based on how δ5 applies to them with respect to v. Let J be the set of indices j ∈ {1, 2, . . . , d (v)} such that δ5 applies to f j with respect to v. The definition of δ5 precludes the possibility that δ5 applies to two consecutive faces around v. Let K be any maximal set of indices k ∈ {1, 2, . . . , d (v)} such that (modulo d (v)) both k − 1 and k + 1 are in J and neither k − 2 nor k + 2 are in K.
To define the triples, each face with index in K ∪ J K is grouped with the two faces neighbouring it around v. Note that by the maximality of K these triples are all pairwise disjoint. If a face f i around v is not in a triple, then by δ6 -δ10 we know that where the computation of indices is modulo d (v) in {1, . . . , d (v)}. Observe that for every index j ∈ K only δ10 may apply to f j with respect to v (by using (4) together with the assumption on ∆ to exclude δ6 -δ8, as well as a brief inspection of δ9), meaning that δ f j (v) is − 3 2 . Moreover, for every index j ∈ J K we see that δ f j−1 (v) and δ f j+1 (v) are both at least − 9 4 , since δ 6 does not apply to f j−1 or f j+1 with respect to v (and the same indeed is also the case for δ 7 ). We conclude for every index j ∈ K ∪ J K that Hence we have in total Claim 25 allows us to finish our analysis of the vertices. If v ∈ T , then γ(v) = −3 and δ(v) − 5 2 ∆ + 5 2 . We see that (5) holds, as ∆ 17.

Proof of Theorem 11
Recall the notation and terminology given in the introduction of this section.
Also this time, if G is not connected, then we extend the edge-colouring one component at a time. The colouring of a component C with ∆(C) 22 can be extended using the results on lines 1 -3 of Table 1. Next, the statement of Theorem 11 is true for graphs with maximum degree 23 and exactly 23 edges. We use induction on E(G), and proceed with the induction step. So we may assume that G is connected and has at least 24 vertices, since ∆ 23. Let M be a precoloured distance-3 matching.
We first observe that Indeed, suppose that the inequality does not hold for some uv / If a vertex v has a (unique) neighbour in V 1 ∪ T 2 , then we always choose v 1 to be this neighbour. In that case f d(v) = f 1 , and that face is called f 1 again. Note that it is possible for other faces to be the same as well (if v is a cut-vertex), but we will not identify those multiple names of the same face.
Proof. Consider the set F of edges in E(G) with one end-vertex in V 2 and the other in V ∆ . Note that F ∩ M = ∅ by the definition of V 2 . The subgraph with vertex set V 2 ∪ V ∆ and edge set F is bipartite; we assert it is acyclic. For suppose there exists an (even) cycle C with E(C) ⊆ F . By induction if ∆(G−E(C)) 23, by Theorem 10 if ∆(G−E(C)) = 22, and by Proposition 23 if ∆(G − E(C)) = 21, we can extend the precolouring of M to G − E(C) using the palette K. But then we can further extend this colouring to the edges of C, since each one sees only ∆ − 2 coloured edges, and even cycles are 2-edge-choosable.
Since each vertex in V 2 is incident with precisely two edges in F , we have |V ∆ | + |V 2 | > |F | = 2|V 2 |. The claim follows.
We use a discharging argument to complete the proof. First, let us assign to each vertex v a charge α1: α(v) = 3d(v) − 6, and to each face f a charge α2: α(f ) = −6.
For each vertex v we define β(v) as follows.
Finally, for each face f and vertex v ∈ W − f we define δ f (v) and δ v (f ) as follows.
Thus, in order to reach a contradiction, it is enough to show that for every vertex v: and that for every face f : Let f be a face. As G is simple, (9) clearly holds. So, by checking δ1 -δ6, we see we only have to consider the case where v ∈ T [3,6] ∪ U [2,5] . (Recall that vertices from V 1 ∪ T 2 do not appear in W − f .) If v ∈ U 2 , then let u and w be the neighbours of v. Consider first the case where both u and w have degree ∆. Then they both belong to V − (f ), so (9) follows, since δ v (f ) = 1 and δ u (f ) Now let v be a vertex. Recall that α(v) = 3d(v) − 6. Furthermore, if v has a neighbour in V 1 ∪ T 2 , then the two consecutive faces incident with that neighbour are counted as one face; all other faces are counted separately. Finally, as noted earlier, a vertex can have at most one neighbour in V 1 ∪ T 2 If d(v) = 1, then α(v) = −3 and γ(v) = 3. Since β(v) = δ(v) = 0, we immediately obtain (8).
Since M is distance-3, none of δ4 and δ5 applies to v, and v is incident 0, and hence (8) is satisfied again. Next assume that v ∈ U , and so γ(v) = 0. The fact that M is distance-3 ensures that δ4 applies to at most one face with respect to v, and δ5 applies to at most two faces with respect to v. Consequently, δ(v) . Combined with the assumption that ∆ 17, this is always enough to satisfy (8).
So we are left with the case where v ∈ U . Since M is a distance-3 matching, at most one of γ2, γ3 applies and at most one of δ4, δ5 applies. Moreover, if γ2 does apply, then γ(v) = −3 and neither δ4 nor δ5 applies. This means that the vertex v is incident with ∆ faces, and for each of those faces f we have δ f (v) = − 5 2 . If γ2 does not apply, then γ(v) −2. The vertex v is incident with ∆ faces, and for ∆ − 1 of those faces f we have δ f (v) = − 5 2 . For the final face f either δ4 or δ5 may apply, so δ f (v) ∈ {−4, −3, − 5 2 }. Using that ∆ 23, we can check that (8) is satisfied in all cases.
This confirms (8) for all vertices and completes the proof of the theorem.

Avoiding Prescribed Colours on a Matching
In this section, we show the following statement, which directly implies Theorem 12. To establish Theorem 27, we use a result mentioned just after Conjecture 3.
Theorem 28 (Berge and Fournier [7]). Let G be a multigraph with maximum degree ∆(G) and maximum multiplicity µ(G), and let M be a matching in G. Then there exists a proper edge-colouring of G using the palette [∆(G) + µ(G)] such that every edge of M receives the same colour.
Proof of Theorem 27. We may assume without loss of generality that M 1 is a maximal matching in G \ M . We set Let ψ be a partial proper edge-colouring of G using colours in [∆(G) + µ(G)] such that (i) ψ(e) = 1 for every e ∈ M 1 ; (ii) ψ(e ) = 1 for every e ∈ M ; (iii) every edge of E(G) \ B receives a colour under ψ; and (iv) the number of edges of B that receive a colour under ψ is maximal.
To show that ψ is well defined, we need to prove the existence of a partial proper edge-colouring of G − B using the palette [∆(G) + µ(G)] that satisfies (i) -(iii).
To this end, let G = G − B. By Theorem 28, there is a proper edge-colouring φ of G using colours in [∆(G) + µ(G)] such that every edge in M 1 receives colour 1. By the definition of B, each edge in M \ B is incident to at least one edge in M 1 . Each edge in M 1 receives colour 1 under φ and therefore φ does not map any edge of M \ B to colour 1. Thus φ ensures that ψ exists.
We now show that every edge of B receives a colour under ψ, which completes the proof. Suppose, on the contrary, that xy ∈ B is an edge that is not coloured by ψ. We start by making the following observations. Indeed, if e is an edge that is coloured 1, then e / ∈ M and e is not adjacent to an edge in M 1 , since all such edges are also coloured 1. Consequently, e ∈ M 1 , as M 1 is a maximal matching of G − M .
Claim 29 and the definition of B ensure the following.
Claim 30. Neither x nor y is incident with an edge that is coloured 1.
be the set of colours that do not appear on edges incident to v. Claim 30 states that A x and A y both contain the colour 1. Indeed, for if v is not incident to an edge in M 1 , then by Claim 30 the edge xv could be added to M 1 to form a larger matching in G − M , thereby contradicting the maximality of M 1 . We know that the edge xy is not yet coloured so both A x and A y must contain some colour different from 1 and we shall from now on redefine A y to be A y \ {1}, which is not empty. We consider the following iterative procedure.
Initially (t = 0), we set D 0 = {y}. At each step t 1, we form the set D t as follows: D i some edge between v and x has its colour in w∈D t−1 A w .
Since i 0 D i ⊆ N G (x) and D i ∩ D j = ∅ if 0 i < j, there exists a least non-negative integer t 0 such that D t 0 +1 = ∅. We define D = i t 0 D i . We consider now two cases.
Case 1. Assume that there exist a vertex w ∈ D and a colour c ∈ A w ∩ A x . Since the subsets D 0 , . . . , D t 0 are pairwise disjoint, there is precisely one integer t 1 such that w ∈ D t 1 . There exists a sequence y = w 0 , w 1 , w 2 , . . . , w t 1 = w of vertices such that w i ∈ D i and (at least) one edge e i between x and w i has a colour in A w i−1 , whenever 1 i t 1 .
For the second case, we need the following two observations. Claim 32. For every z ∈ N G (x), it holds that µ(G) |A z |.
The only case that is not trivial is when z = y, due to our redefinition of A y . However, as the edge xy is not coloured, the vertex y sees at most ∆(G) − 1 different colours, which implies the statement.
Let H be the bipartite subgraph of G induced by the bipartition ({x}, D). (In particular, the edges of G between vertices in D are not in H.) The next statement follows directly from the fact that the number of coloured edges between x and y is at most µ(G) − 1.
We can now proceed with the second case.
Case 2. For every vertex w ∈ D and every colour c ∈ A w , there exists an edge e w between x and a vertex z ∈ D such that ψ(e w ) = c. By Claims 32 and 33, we know the electronic journal of combinatorics 25(3) (2018), #P3.1 that the number of colours appearing in the bipartite graph H is less than |D| · µ(G), which is at most w∈D |A w |. This implies that there are two distinct vertices v 1 and v 2 in D ⊆ N G (x) with A v 1 ∩ A v 2 = ∅. Let c 1 ∈ A v 1 ∩ A v 2 and note that c 1 = 1 by Claim 31. Let c 2 ∈ A x \ {1}. Then c 2 / ∈ A v 1 ∪ A v 2 and c 1 / ∈ A x . (And hence c 1 = c 2 .) For i ∈ {1, 2}, let P i be the maximal alternating path with colours c 1 and c 2 beginning at v i . Note that x cannot belong to both paths. But if x does not belong to P i , then we may swap c 1 and c 2 along the edges of P i . This leads us back to Case 1 because then c 2 belongs to A x ∩ A v i . (Note that such a swap affects neither the colours of the edges inside H nor those of edges in M 1 .) We have shown that in each case there exists a partial proper edge-colouring using colours in [∆(G) + µ(G)] and satisfying (i) -(iii) that assigns colours to more edges of B than ψ does, a contradiction.

Conclusion
During the preparation of this manuscript, we learned of a related work in the context of graph limits [13], in which is proposed the following conjecture that has a similar flavour to our Conjecture 3.
Conjecture 34 (Csóka, Lippner and Pikhurko [13]). Let G be a graph such that every vertex is of degree at most d, except one of degree d + 1. Using the palette K = [d + 1], suppose that at most d − 1 pendant edges are precoloured. This precolouring can be extended to a proper edge-colouring of all of G.
The authors of Conjecture 34 proved the weaker statement with K = [d + 9 √ d] instead of K = [d + 1].
With respect to Question 1, rather than imposing conditions on the matching M , we could instead constrain the precolouring. In the light of Theorem 14 and the result of Berge and Fournier [7], the following is a natural strengthened version of Conjecture 3.

Conjecture 35.
Let G be a multigraph with maximum degree ∆(G) and maximum multiplicity µ(G). Using the palette K = [∆(G) + µ(G)], any precoloured matching such that no two edges precoloured differently are within distance 2 can be extended to a proper edge-colouring of all of G.
We may rephrase Theorem 12 in the language of list colouring as follows: for any multigraph G, any matching M in G, and any list assignment L : E(G)  It would also be interesting if either of Conjectures 35 and 36 could be confirmed with the constant 2 replaced by any larger fixed integer.