On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star chromatic index χ ′ st ( G ) of G is the minimum number t for which G has a star edge coloring with t colors. We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b . Let k be an integer ( k ≥ 1); we prove that if b = 2 k + 1, then χ ′ st ( G ) ≤ 3 k + 2; and if b = 2 k , then χ ′ st ( G ) ≤ 3 k ; both upper bounds are sharp. We also consider complete bipartite graphs; in particular we determine the star chromatic index of such graphs when one part has size at most 3, and prove upper bounds for the general case. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 6; in particular we settle this conjecture for cubic Halin graphs. © 2021TheAuthors


Introduction
A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four.The star chromatic index χ ′ st (G) of G is the minimum number t for which G has a star edge coloring with t colors.Star edge coloring was recently introduced by Liu and Deng [14], motivated by the vertex coloring version, see e.g.[1,5].This notion is intermediate between acyclic edge coloring, where every two-colored subgraph must be acyclic, and strong edge coloring, where every color class is an induced matching.
Dvořák et al. [4] studied star edge colorings of complete graphs and obtained the currently best upper and lower bounds for the star chromatic index of such graphs.A fundamental open question here is to determine whether χ ′ st (K n ) is linear in n.Bezegová et al. [2] investigated star edge colorings of trees and outerplanar graphs.Lei et al. [11] proved that it is NP-complete to determine if a graph G satisfies that χ ′ st (G) ≤ 3. Wang et al. [19,20] quite recently obtained some upper bounds on the star chromatic index of graphs with maximum degree four, and also for some families of planar and related classes of graphs.Some further results on star edge colorings of subcubic (i.e. with maximum degree at most three) and sparse graphs appear in [6,[8][9][10]12,15].Besides these results, very little is known about star edge colorings.
In this paper, we primarily consider star edge colorings of bipartite graphs.As for complete graphs, a fundamental problem for complete bipartite graphs is to determine whether the star chromatic index is a linear function on the number of vertices.We determine the star chromatic index of complete bipartite graphs where one part has size at most 3, and obtain some bounds on the star chromatic index for larger complete bipartite graphs.Note that the complete bipartite graph K r,s requires exactly rs colors for a strong edge coloring; indeed it has been conjectured [3] that any bipartite graph where the parts have maximum degrees ∆ 1 and ∆ 2 , respectively, has a strong edge coloring with ∆ 1 ∆ 2 colors.As we shall see, for star edge colorings the situation is quite different.
Furthermore, we study star chromatic index of bipartite graphs where the vertices in one part all have small degrees.Nakprasit [17] proved that if G is a bipartite graph where the maximum degree of one part is 2, then G has a strong edge coloring with 2∆(G) colors.Here we obtain analogous results for star edge colorings: we obtain a sharp upper bound for the star chromatic index of a bipartite graph where one part has maximum degree two.
Finally, we consider the following conjecture first posed in [4].In this paper we verify that the conjecture holds for some families of graphs with maximum degree three, namely bipartite graphs where one part has maximum degree 2, cubic Halin graphs and another family of planar graphs.

Bipartite graphs
In this section we consider star edge colorings of bipartite graphs.For an edge coloring f of a graph G and a vertex u of G, we shall denote by f (u) the set of colors of all edges incident with u.
We first consider complete bipartite graphs.Trivially χ ′ st (K To prove the upper bound, we give an explicit star edge coloring f of K 2,d .We set f (x 1 y i ) = i, for i = 1, . . ., d, and The coloring f is a star edge coloring using exactly 2d − Wang et al. [20] proved that χ ′ st (K 3,4 ) = 7, and it is known that χ ′ st (K 3,3 ) = 6 [4].Using Proposition 2.1, we can prove the following.

Theorem 2.2. For the complete bipartite graph K
Proof.Let us first consider the case when d is even.The lower bound 3d 2 follows immediately from Proposition 2.1, so let us turn to the proof of the upper bound.We shall give an explicit star edge coloring of K 3,d using 3d 2 colors.
Let X and Y be the parts of K 3,d , where We define a star edge coloring f by setting and Clearly, f is a proper edge coloring of K 3,2k with 3k colors.Suppose that K 3,2k contains a 2-edge-colored path or cycle F with four edges.Let us prove that F does not contain two edges e 1 and e 2 incident with the same vertex from X , say x 1 , and two other edges e 3 and e 4 incident with another vertex from X , say x 2 .Then, since the restriction f ′ of f to the subgraph induced by X ∪ U satisfies f ′ (x 1 ) ∩ f ′ (x 2 ) = ∅ (and similarly for the subgraph induced by X ∪ V ), we may assume that f (e 1 ) ∈ {1, . . ., k} and f (e 2 ) ∈ {k+1, . . ., 2k}.However, no edge incident with x 2 is colored by a color from {1, . . ., k}, which contradicts that F is 2-edge-colored.Suppose now that there is a 2-edge-colored path F on 4 edges, where exactly two edges are incident to the same vertex from X , say x 2 .Then, as before, we may assume that the two edges e 2 and e 3 of F that are incident with x 2 satisfy that f (e 2 ) ∈ {2k + 1, . . ., 3k} and f (e 3 ) ∈ {k + 1, . . ., 2k}.This means that the edge e 1 of F that is incident with x 1 must satisfy f (e 1 ) ∈ {k+1, . . ., 2k}, and so, f (e 1 ) = f (e 3 ).Thus, for the edge e 4 of F incident with x 3 it holds that f (e 4 ) ∈ {2k+1, . . ., 3k} and f (e 4 ) = f (e 2 ).However, by the construction of f we have that f (e 4 ) = f (e 2 )−1 (except if f (e 2 ) = 2k+1, which implies that f (e 4 ) = 3k); this contradicts that F is 2-edge-colored.
Let us now consider the case when d is odd; suppose d = 2k + 1.The upper bound follows immediately from the even case, since K 3,2k+1 is a subgraph of K 3,2k+2 .Let us prove the lower bound.Let X and Y be the parts of K 3,d , where X = {x 1 , x 2 , x 3 } and consider a star edge coloring of K 3,d .Since the subgraph induced by {x 1 , x 2 } ∪ Y is isomorphic to K 2,d , there are at most k colors which can appear at both x 1 and x 2 .Since the same holds for x 1 and x 3 , and x 2 and x 3 , it follows that we need at least 3k + 3 colors for a star edge coloring of K 3,d .□ Next, we consider complete bipartite graphs where the parts have size at least 4. Let us first establish a lower bound on the star chromatic index of such graphs.

Proposition 2.3. For the complete bipartite graph
Proof.Let X and Y be the parts of K 4,d , where X = {x 1 , x 2 , x 3 , x 4 }, and let C i be the set of colors used on the edges incident with the vertex x i .From the argument in the proof of Proposition 2.1, we can conclude that the sets C i may overlap in at most d/2 colors.Thus the optimization problem minimize

⌋
. The proposition follows easily by decomposing K 4,d into s copies H 1 , . . ., H s of K 4,12 and possibly one copy J of a complete bipartite graph K 4,a , where 1 ≤ a ≤ 11, and then using disjoint sets of colors for star edge colorings of each of the graphs H 1 , . . ., H s and J; these star edge colorings together form a star edge coloring of K 4,d .□ Note that it follows from Proposition 2.3 that the upper bound in the preceding proposition is in fact sharp for an infinite number of values of d.
Using computer searches we have also determined the star chromatic index for some additional complete bipartite graphs; see Tables 2 and 3. Again, explicit colorings appear in Appendix B.Moreover, using a similar linear program technique as in the proof of Proposition 2.3, one can prove lower bounds on the star chromatic index for further families of complete bipartite graphs.Let us here just list a few cases corresponding to the values in the tables above: Finally, let us note some further consequences of the above results for general complete bipartite graphs.By decomposing a general complete bipartite graph K r,d into complete bipartite graphs where one part has size e.g. at most 3, and using disjoint sets of colors for star edge colorings of distinct complete bipartite subgraphs, we deduce, using . However, using the values of star chromatic indices in Tables 2 and 3, it is possible to deduce an upper bound on χ ′ st (K r,d ) which is better for large values of r and d.
Note that Dvořák et al. [4] obtained an asymptotically better bound: it follows from their results that for every ε > 0, there is a constant C > 0, such that for every Next, we turn to general bipartite graphs with restrictions on the vertex degrees.Our first task is to generalize Proposition 2.1 to general bipartite graphs with even maximum degree.
In the following we use the notation G = (X , Y ; E) for a bipartite graph G with parts X and Y and edge set E = E(G).We denote by ∆(X) and ∆(Y ) the maximum degrees of the vertices in the parts X and Y , respectively.A bipartite graph G = (X , Y ; E) where all vertices in X have degree 2 and all vertices in Y have degree d is called (2, d)-biregular.
If the vertices in one part of a bipartite graph G has maximum degree 1, then trivially χ ′ st (G) = ∆(G).For the case when the vertices in one of the parts have maximum degree two, we have the following.
Note that the upper bound in Theorem 2.6 is sharp, as follows from Proposition 2.1.
For the proof of this theorem we shall use the following lemma.
Lemma 2.7.If G is (2, 2k)-biregular with parts X and Y , then it decomposes into subgraphs F i such that d F i (x) ∈ {0, 2} for every x ∈ X and d F i (y) = 2 for every y ∈ Y .
Proof.From a (2, 2k)-biregular graph G, construct a 2k-regular multigraph H by replacing every path of length 2 with an internal vertex of degree two by a single edge.By Petersen's 2-factor theorem [18], H has a decomposition into 2-factors; these 2-factors induce the required subgraphs of G. □ We shall also use the simple fact that every even cycle has a star edge coloring with three colors, the proof of which is left to the reader.
Proof of Theorem 2.6.If G = (X , Y ; E) is not (2, 2k)-biregular, then it is a subgraph of such a graph, so it suffices to consider the case when G is (2, 2k)-biregular.
Assume, consequently, that G is a (2, 2k)-biregular graph.By the preceding lemma, G decomposes into subgraphs F 1 , . . ., F k such that d F i (x) ∈ {0, 2} for every x ∈ X and d F i (y) = 2 for every y ∈ Y .Since each F i is a collection of even cycles, it has a star edge coloring with three colors.
For i = 1, . . ., k, we color each F i with colors 3i−2, 3i−1, 3i so that each F i gets a star edge coloring with 3 colors.This yields a star edge coloring f of G; indeed, for every vertex x of X , all colors on edges incident with X is in {3i−2, 3i −1, 3i} for some i.Hence, if there is a 2-colored cycle or path J with four edges in G, then since J contains at least two vertices from X , it must be colored by two colors from {3i − 2, 3i − 1, 3i} for some i.This implies that all edges of J are in F i , which contradicts that the restriction of f to each F i is a star edge coloring.We conclude that f is in fact a star edge coloring of G. □ For the case d = 3, we can generalize Proposition 2.1 as follows.
To prove the theorem, we shall use the following easy lemma from [15], and the notion of a list star edge coloring.A list assignment L for a graph G is a map which assigns to each edge e of G a set L(e) of colors.If each of the lists has size k, we call L a k-list assignment.If G admits a star edge coloring ϕ such that ϕ(e) ∈ L(e) for every edge e of G, then G is star L-edge-colorable; ϕ is a star L-edge coloring of G.The graph G is star k-edge-choosable if it is star L-edge-colorable for every list assignment L, where |L(e)| ≥ k for every e ∈ E(G).Lemma 2.9.If C is any cycle distinct from C 5 , then C is star 3-edge-choosable.
By the distance between two edges e and e ′ of a graph, we mean the smallest number of edges in a path from an endpoint of e to an endpoint of e ′ .
Before proving Theorem 2.8, we have to notice that for C 4 -free graphs satisfying the condition in Theorem 2.8, this result can be deduced from the result of [16].In [16], it is proved that the incidence chromatic number of a subcubic graph is at most 5; the incidence chromatic number of a graph G is equal to the strong chromatic index of G * , where G * is the graph obtained from G by subdividing each edge of G.When G is a cubic graph (with no multiple edges), the graph G * is a (2, 3)-biregular graph with no cycles of length four.Now, since the strong chromatic index of a graph is an upper bound for its star chromatic index, the result follows.
Since the result of [16] only applies to C 4 -free (2, 3)-biregular graphs, and to have a self-contained paper, we give our short proof of Theorem 2.8.
Proof of Theorem 2.8.Since any cycle of even length has a star 3-edge coloring, we may assume that G has maximum degree 3.
Assume that G is a counterexample to the theorem which minimizes |V (G)| + |E(G)|.Then G satisfies the following: (i) G is connected; (ii) G does not contain any vertex of degree 1; (iii) no two vertices of degree 3 are adjacent; (iv) G does not contain two vertices of degree 3 that are linked by a path P of length at least four, where all internal vertices of P have degree 2. Thus any vertex of degree 2 has two neighbors of degree 3, so G is (2, 3)-biregular.
Statements (i)-(iii) are straightforward.To see (iv), assume that P is such a path, and let u 1 u 2 and u 2 u 3 be two adjacent edges of P, where u 1 , u 2 and u 3 all have degree 2. By assumption G − u 2 has a star edge coloring with 5 colors.Now we can color u 1 u 2 with a color not appearing on an edge of distance at most 1 from u 1 u 2 in G; there are at most four such edges, so this is possible.Next, we can color u 2 u 3 by a color not appearing on u 1 u 2 or on any edge at distance at most 1 from u 2 u 3 in G − u 1 u 2 ; there are at most four such edges, so we can pick a color from {1, 2, 3, 4, 5} for u 2 u 3 .This yields a star 5-edge coloring of G; a contradiction, and so, (iv) holds.
Let C 2k = u 1 u 2 . . .u 2k u 1 be a shortest cycle of G; if G does not have a cycle, then it has a vertex of degree 1 and thus violates condition (ii), a contradiction.Then the graph H = G − E(C 2k ) has a star 5-edge coloring.
We define a new star edge coloring f of H by recoloring every pendant edge e of H by a color from {1, 2, 3, 4, 5} not appearing on any edge of distance at most 1 from e; there are at most three such edges, so this is possible.
Next, we define a list assignment L for C 2k with colors from {1, 2, 3, 4, 5} by for each edge u i u i+1 of C 2k forbidding the colors on the two pendant edges of H with smallest distance to u i u i+1 in G. Then every edge of C 2k receives a list of size at least 3; so by Lemma 2.9 it has star L-edge coloring.This coloring along with the star edge coloring f of H form a star edge coloring of G with 5 colors.□ Note that the preceding theorem settles a particular case of Conjecture 1.1.
Let us briefly remark that there are (2, 3)-biregular graphs with χ ′ st (G) = 4; while such examples with χ ′ st (G) = 3 trivially do not exist.Take two copies of P 5 , and denote these copies by the resulting graph G is (2, 3)-biregular.We define a proper edge coloring f of this graph by setting and and by coloring all edges in E ′ by color 4. The coloring f is a star edge coloring, because all edges of E ′ are adjacent to edges of three distinct colors.
For bipartite graphs G = (X , Y ; E) with ∆(X) = 2 and odd maximum degree at least 5, we can use Theorem 2.8 for proving the following.
Note that by Proposition 2.1, Theorem 2.10 is sharp.
For the proof of Theorem 2.10 we shall need the following theorem due to Bäbler, see e.g.[7].
If the graph G is obtained from H by subdividing every edge of H, then we say that H is the condensed version of G.
The proof of Theorem 2.10 is similar to the proof of the main result of [7]; hence we omit some details.
Proof of Theorem 2.10.Since every graph satisfying the conditions in the theorem is a subgraph of a (2, 2k+1)-biregular graph, it suffices to prove the theorem for (2, 2k + 1)-biregular graphs.The proof is by induction on k.The case k = 0 is trivial, and Theorem 2.8 settles the case k = 1.Now assume that k ≥ 2 and that G is a (2, 2k + 1)-biregular graph.Let H be the condensed version of G; then H is (2k + 1)-regular.
If H has at most one bridge, then by Theorem 2.11, H has a 2-factor.In G, this 2-factor corresponds to a subgraph F , where all vertices of Y have degree 2, and every vertex of X has degree 2 or 0. Note that the graph G ′ obtained from G − E(F ) by removing all isolated vertices is a (2, 2k − 1)-biregular graph.By the induction hypothesis, G ′ has a star edge coloring with 3(k − 1) + 2 colors.By star edge coloring all cycles of F with 3 additional colors, we obtain a star edge coloring with 3k + 2 colors of G. Now assume that H has at least two bridges.We proceed as in [7]: Let B 1 , . . ., B r be the maximal bridgeless connected subgraphs obtained from H by removing all bridges.For each subgraph B i we construct a (2k + 1)-regular multigraph containing B i by proceeding as follows: If there is an even number of bridges in H with endpoints in B i , then we add a number of copies of the graph A consisting of 2k parallell edges, the endpoints of which we join to endpoints in B i of removed bridges by a single edge, respectively; if there is an odd number of bridges with endpoints in B i , then we also add a subgraph T consisting of a triangle xyzx where x and y are joined by k + 1 parallell edges, x and z are joined by k parallell edges, and y and z are joined by k parallell edges, and z is joined by an edge to one endpoint in B i of a removed bridge.This yields a (2k+1)-regular multigraph J i containing B i .We set Note that in J a bridge b of H is replaced by two edges joining the endpoints of b with vertices of subgraphs isomorphic to A or T .
By Theorem 2.11, J has a 2-factor.Thus, by proceeding as in the preceding case, we may construct a star edge coloring f with 3k+2 colors of the corresponding (2, 2k+1)-biregular graph D obtained from J by subdividing all edges of J. Now, let K be the graph obtained from D by removing all edges of D that are in subgraphs that correspond to the added subgraphs in J that are isomorphic to A or T , and thereafter removing all isolated vertices.The obtained graph K is identical to the graph obtained from B = B 1 ∪ • • • ∪ B r by (i) subdividing all edges of B, and (ii) for every bridge uv of H adding a path uu 2 u 3 with origin at u, where u 2 , u 3 are new vertices and u 3 has degree 1, and adding a path vv 2 v 3 with origin at v, where v 2 , v 3 are new vertices and v 3 has degree 1.
Thus each path of length 2 in G that corresponds to a bridge in H is represented by two distinct paths of length 2 in K ; moreover, if we identify every pair of such paths corresponding to the same bridge in H, then we obtain a graph isomorphic to G.
Let f K be the restriction of f to K ; this is a star edge coloring of K with 3k + 2 colors.We recolor every pendant edge of K by a color from {1, . . ., 3k + 2} which does not appear on an edge of distance at most 1 from the pendant edge; the obtained coloring f ′ K is a star edge coloring with the property that no pendant edge is in a bicolored path of length at least 3. Now, to obtain a star edge coloring of G from f ′ K we may successively ''paste'' together components of K by identifying paths that correspond to the same bridge in H and permuting colors in one of the components so that the colorings agree on the identified paths; this ''pasting process'' can be done exactly as in [7] (e.g. by doing a Depth-First-Search in the tree with vertices for the subgraphs B i and edges for the bridges of H), so we omit the exact details here.Since in K , any bicolored path with a pendant edge has length at most 2, this yields a star edge coloring of G. □

Planar cubic graphs
As mentioned above, Conjecture 1.1 has been verified for outerplanar graphs.A particularly interesting special case of Conjecture 1.1 is planar graphs; this particular case is still wide open.In this section we provide two results in this direction.
Recall that a Halin graph is a planar graph constructed from a planar drawing of a tree with at least four vertices and with no vertices of degree two by connecting its leaves by a cycle that crosses none of its edges.We shall first prove that cubic Halin graphs have star chromatic index at most 6.Note that this upper bound is sharp, since the complement of a 6-cycle is a cubic Halin graph attaining this bound (see e.g.[15]).Our proof is similar to the proof in [13] of the fact that cubic Halin graphs have strong chromatic index at most 7.
Proof.Let G = T ∪C, where T is a tree and C is an adjoint cycle containing all pendant vertices of T .Our proof proceeds by induction on the length m of the cycle C .It is straightforward that every cubic Halin graph with m ≤ 5 has star chromatic index at most 6; indeed in such a graph G = T ∪ C , the tree T consists of a path P = u 0 u 1 . . .u k along with the vertices So let us assume that m ≥ 6.Let P = u 0 u 1 . . .u l be a path of maximum length in T .Since ∆(T ) ≤ 3 and m ≥ 6, l ≥ 5.Moreover, since P is maximum, all neighbors of u 1 , except u 2 , are leaves.We set w = u 3 , u = u 2 , v = u 1 .Moreover, let v 1 and v 2 be the neighbors of v on C and label some other vertices in G according to Fig. 1.
Since d G (u) = 3, there is a path Q from u to x 1 or y 1 with V (P) ∩ V (Q ) = {u}.Suppose without loss of generality that there is such a path from u to y 1 .Then, since P is a path of maximum length in T , Q has length at most two; that is, uy 3 ∈ E(T ) or u = y 3 .If the former holds, then y 3 y 2 ∈ E(T ), and in the latter case, uy 1 ∈ E(T ).
This yields a star edge coloring of G.

Fig. 1 .
Fig. 1.The subgraph of G used in the induction step.

Fig.
Fig. A star edge coloring of K 5,8 with 15 colors.
1,d ) = d, and it is straightforward that χ ′ st (K 2,2 ) = 3.For general complete bipartite graphs where one part has size 2, we have the following easy observation.If x 1 and x 2 have at least ⌊d/2⌋ + 1 common colors on their incident edges, say 1, . . ., ⌊d/2⌋ + 1, then there is at least one vertex in Y which is incident with two edges both of which have colors from {1, . . ., ⌊d/2⌋ + 1}; this implies that there is a 2-colored P 4 or C 4 in K 2,d .Hence, there are at least 2d − ⌊d/2⌋ distinct colors in a star edge coloring of K 2,d .
).A linear integer program description of this problem is given in Appendix A. Solving the linear relaxation of this problem gives the desired lower bound.□Ford≤12, the values of χ ′ st (K 4,d ) are given in Table 1.Explicit colorings realizing these values are given in Appendix B,and for all values of d except d = 6 they can be proved optimal by solving the optimization problem in the proof of Proposition 2.3.For the case d = 6, a computer search showing that the value in Table1is optimal has been conducted.The fact that χ ′ st (K 4,12 ) = 20 can be used to derive a general upper bound on the star chromatic index of complete bipartite graphs with four vertices in one part.