Hadwiger's conjecture for 3-arc graphs

The 3-arc graph of a digraph $D$ is defined to have vertices the arcs of $D$ such that two arcs $uv, xy$ are adjacent if and only if $uv$ and $xy$ are distinct arcs of $D$ with $v\ne x$, $y\ne u$ and $u,x$ adjacent. We prove that Hadwiger's conjecture holds for 3-arc graphs.


Introduction
A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. An H-minor is a minor isomorphic to H. The Hadwiger number h(G) of G is the maximum integer k such that G contains a K k -minor, where K k is the complete graph with k vertices.
In 1943, Hadwiger [10] posed the following conjecture, which is thought to be one of the most difficult and beautiful problems in graph theory: Hadwiger's Conjecture. For every graph G, h(G) ≥ χ(G).
In this paper we prove Hadwiger's conjecture for a large family of graphs. Such graphs are defined by means of a graph operator, called the 3-arc graph construction (see Definition 1), which bears some similarities with the line graph operator. This construction was first introduced by Li, Praeger and Zhou [15] in the study of a family of arc-transitive graphs whose automorphism group contains a subgroup acting imprimitively on the vertex set. (A graph is arc-transitive if its automorphism group is transitive on the set of oriented edges.) It was used in classifying or characterizing certain families of arc-transitive graphs [9,11,15,17,23,24,25]. Recently, various graph-theoretic properties of 3-arc graphs have been investigated [1,12,13,22].
The original 3-arc graph construction [15] was defined for a finite, undirected and loopless graph G = (V (G), E(G)). In G, an arc is an ordered pair of adjacent vertices. Denote by A(G) the set of arcs of G. For adjacent vertices u, v of G, we use uv to denote the arc from u to v, and {u, v} the edge between u and v. We emphasise that each edge of G gives rise to two arcs in A(G). A 3-arc of G is a 4-tuple of vertices (v, u, x, y), possibly with v = y, such that both (v, u, x) and (u, x, y) are paths of G. The 3-arc graph of G is defined as follows: Definition 1. [15,24] Let G be an undirected graph. The 3-arc graph of G, denoted by X(G), has vertex set A(G) such that two vertices corresponding to arcs uv and xy are adjacent if and only if (v, u, x, y) is a 3-arc of G.
The 3-arc graph construction can be generalised for a digraph D = (V (D), A(D)) as follows [12], where A(D) is a multiset of ordered pairs (namely, arcs) of distinct vertices of V (D). Here a digraph allows parallel arcs but not loops. Let D be the digraph obtained from an undirected graph G by replacing each edge {x, y} by two opposite arcs xy and yx. Then, X(D) = X(G).
Knor, Xu and Zhou [12] introduced the notion of 3-arc colouring a digraph, which can be defined as a proper vertex-colouring of X(D). The minimum number of colours in a 3arc colouring of D is called the 3-arc chromatic index of D, and is denoted by χ ′ 3 (D). Then χ(X(D)) = χ ′ 3 (D). The main result of this paper is the following: Theorem 1. Let D be a digraph without loops. Then h(X(D)) ≥ χ(X(D)).
Note that in the case of the 3-arc graph of an undirected graph, we have obtained a much simpler proof of Theorem 1.

Preliminaries
We need the following notation. Let D = (V (D), A(D)) be a digraph. We denote by A D {x, y} the set of arcs between vertices x and y, and by A D (x) the set of arcs outgoing from x. Then vertices x and y are adjacent if and only if A D {x, y} = ∅. When |A D {x, y}| = 1, we misuse the notation A D {x, y} to indicate the arc between x and y. An in-neighbour (respectively, outneighbour) of a vertex x of D is a vertex y such that yx ∈ A(D) (respectively, xy ∈ A(D)). The set of all in-neighbours (respectively, out-neighbours) of x is denoted by for all distinct vertices x and y of D. A tournament is a simple digraph whose underlying undirected graph is complete.
For an undirected graph G, the degree of a vertex v in G is denoted by d G (v), and the minimum degree of G is denoted by ( . G). We omit the subscript when there is no ambiguity.
A K t -minor in G can be thought of as t connected subgraphs in G that are pairwise disjoint such that there is at least one edge of G between each pair of subgraphs. Each such subgraph is called a branch set. Lemma 2. Let D be a tournament on n ≥ 5 vertices. Then h(X(D)) ≥ n.
Proof. Since D is a tournament, A{x, y} is interpreted as a single arc. Denote V (D) = {x, v 0 , v 1 , . . . , v n−2 }. We now construct a collection of n branch sets. For 0 ≤ i ≤ n − 2, where all subscripts are taken modulo n − 1. Clearly, these branch sets are pairwise disjoint. Now we show that each branch set is connected. Note that each B i induces K 2 in X(D).
, U induces a subgraph that contains an (n − 1)-cycle passing through each element of U .
Next we show that these branch sets are pairwise adjacent. For each pair of distinct B i , B j , if j = i + 1 and j = i + 2, then B i and B j are adjacent since A{v i+1 , v i+2 } is adjacent to Note that each feasible arc xy is adjacent in X(D) to each arc in A except vx, and each compatible arc xy is adjacent to each arc in A.
Let A f be an A-feasible set, and A c be an A-compatible set. An (A, A f , A c )-net of size p is a K p -minor in X(D) using only arcs in A ∪ A f ∪ A c such that p := |A| and each branch set has exactly one arc in A. An (A, A f , A c )-net is called a net at v if v is the common tail of all arcs in A. It may happen that one of A f and A c is empty. The following lemma provides some sufficient conditions for the existence of an (A, A f , A c )-net. (1) p = 1; (2) |A c | ≥ 1 and p = 2; (3) |A f | = 3 and p = 3; (4) |A f | ≥ 1 and |A c | ≥ 1 and p = 3; (5) |A c | ≥ 2 and p = 3; Proof. Denote A = {vv 0 , vv 1 , . . . , vv p−1 }, and without loss of generality, assume that (3) |A f | = 3 and p = 3: (4) |A f | ≥ 1 and |A c | ≥ 1 and p = 3: Let ww ′ be an A-compatible arc and A c := {ww ′ }. Note that ww ′ is adjacent to each vv i , and v 0 v ′ 0 is adjacent to vv 2 in X(D). So {vv 0 , ww ′ }, {vv 1 , v 0 v ′ 0 } and {vv 2 } form an (A, A f , A c )-net of size 3. (5) |A c | ≥ 2 and p = 3: Similar to case (4), {vv 0 , ww ′ }, {vv 1 , yy ′ } and {vv 2 } form an (A, A f , A c )-net of size 3, where A c contains two A-compatible arcs yy ′ and ww ′ .
Note that if D contains an (A, A f , A c )-net, then X(D) contains a K p -minor and h(X(D)) ≥ p.
A graph G with chromatic number k is called k-critical if χ(H) < χ(G) for every proper subgraph H of G. The following result is well known:  (b) no vertex-cut of G induces a clique when k ≥ 3 and G is noncomplete [8].

Proof of Theorem 1
In this proof, we assume that, for every pair of distinct vertices u and v of D, there is at most one arc from u to v and at most one arc from v to u. That is, A D {u, v} ⊆ {uv, vu}. That is because all the arcs from u to v can be assigned the same colour and deleting arcs does not increase h(X(D)).
Let D be a digraph.
Note that if uv is redundant then so is vu if it exists. Let D ′ be the digraph obtained from D by deleting all redundant arcs. Let G be the (simple) underlying undirected graph of D ′ . We have the following claim: We now show that f is a 3-arc colouring of D. For every pair of adjacent arcs uv, xy ∈ A(D), we have that A D {u, x} = ∅ (that is, u, x are adjacent), and both uv and xy are not in A D {u, x}. Thus, some arc between u and x is not redundant, and u and x are adjacent Hadwiger's conjecture is true for k-chromatic graphs with k ≤ 6. So assume that χ(X(D)) ≥ 7. Let k := χ(G) and let H be a k-critical subgraph of G. By Lemma 4(a), δ(H) ≥ k − 1.
Let F be an orientation of H such that each arc uv of F inherits the orientation of an arc in A D {u, v} and the number of out-degree 1 vertices in F is minimized. An arc xy ∈ A(D) is called potential if xy / ∈ A(F ). In particular, every redundant arc is potential. F has the following property: Since vw is the unique outgoing arc from v in F , vz is potential. Suppose that z ∈ V (F ). Suppose first that v and z are not adjacent in F . Then each arc between v and z in D including vz is redundant. Since vw ∈ A(D), A D (z) ⊆ A D {z, v}. That is, no arc is outgoing at z in D except possibly zv. Thus, d + F (z) = 0 as desired. Suppose next that v and z are adjacent in F . By the assumption that d + = 2 and the out-degree of every other vertex remains unchanged. Hence F ′ is an orientation of H with less out-degree 1 vertices than F , which is a contradiction.
In addition, for each arc xy of F , by the definition of That is, there is an arc other than yx outgoing from y (hence, d + D (y) ≥ 1) and there is a directed path in D of length 2 starting from the arc xy, even if d + F (y) = 0. Note that F is a simple digraph and and let A f be a maximal A-feasible set. Then |A f | = k ≥ 6 since there exists a directed path of length 2 starting from every arc of A. By Lemma 3(6) with p = k and q = 0, there exists an (A, A f , ∅)-net of size k. Thus, h(X(D)) ≥ k, and the result holds. Now assume that ∆ + (F ) ≤ k − 1. By Lemma 5 and since F has minimum degree at least k − 1, . Let W be the set of all special vertices of F , and let Denote by Q the set of sinks of F . Then each arc of W + has its tail in W and head in Q. Note that W is independent in F , and W ∩ Q = ∅. By Lemma 5, .
. , x r }, and N 2 = {y 1 , y 2 , . . . , y s }. Since F has minimum degree at least k − 1, both r and s are at least k − 2.
Since the arc Thus, for each x l ∈ N 1 , to arc A F {u, x l } ∈ A(F ) we can associate an arc, denoted ϕ(u, x l ), which is chosen from Choose these arcs ϕ(u, x l ) and ϕ(v, y l ) such that if Σ := ∪ r l=1 ϕ(u, x l ) and Π := ∪ s l=1 ϕ(v, y l ) then t := |Σ ∩ Π| is minimized. We now prove that, for each ww ′ ∈ Σ ∩ Π, ww ′ is the unique arc outgoing from w in D, Suppose that |A D (w)| ≥ 2, and ww ′′ is an arc outgoing from w other than ww ′ in D. Then at least one of u and v, say u, is not equal to w ′′ . Now set ϕ(u, w) := ww ′′ and keep ϕ(v, w) = ww ′ . Then |Σ ∩ Π| is decreased. Thus, ww ′ is the unique arc outgoing from w in D.
Then w l ∈ T for each l ∈ [1, t] and t ≤ |T | ≤ min{i, j}. Consider the following cases: In this case, we construct an (A, Since each branch set in A contains an outgoing arc at u other than uv, and each branch set in B contains an outgoing arc at v other than vu, each branch set in A is adjacent in X(D) to each branch set in Each branch set in A contains an outgoing arc at u other than uv, and each branch set in B contains an outgoing arc at v other than vu. Thus each branch set in A is adjacent in X(D) to each branch set in B. Since Thus, no branch set in A intersects a branch set in B. Hence A ∪ B is a K k -minor in X(D).
By symmetry and since uv is not used in this case, if Π − Σ = ∅, then we obtain a K k -minor in X(D). Now assume that Σ = Π. Then |Σ| = |Π| = t = k − 2. Set w 0 := v and w ′ 0 := w 1 . For 0 ≤ l ≤ t, let B l := {uw l , w l+1 w ′ l+1 }, where subscripts are taken modulo t + 1; and let B t+1 := {vw 2 }. For 0 ≤ l < l ′ ≤ t, either uw l is adjacent to w l ′ +1 w ′ l ′ +1 or uw l ′ is adjacent to w l+1 w ′ l+1 . Thus B l is adjacent to B l ′ . Note that vw 2 ∈ B t+1 is adjacent to w 1 w ′ 1 ∈ B 0 and uw l ∈ B l with 1 ≤ l ≤ t. Thus B t+1 is adjacent to every B l with 0 ≤ l ≤ t. Therefore, B 0 , B 1 , . . . , B t+1 form the t + 2 = k branch sets of a K k -minor in X(D).
Note that each arc in Σ is either A-feasible or A-compatible, and no two arcs in Σ share a tail. Let A f (A c , respectively) be the set of A-feasible (A-compatible, respectively) arcs in Σ. Suppose that t ≥ 1 and j = k − 2. If t = 1, then let A be a subset of A F (u) − {uv} with uw 1 ∈ A and |A| = 2. Note that |Σ − Π| = r − t ≥ k − 3 ≥ 3. Then at least one arc in Σ − Π is Acompatible. If t ≥ 2, then let A := {uw 1 , uw 2 }.
If i ≥ t+1, then j ′ = k −i ≤ k −t−1. Let B := {vv 1 , vv 2 , . . . , vv j ′ } be a subset of A F (v) with vw 1 ∈ B. By the assumption that j ≤ k − 3, there is at least one incoming arc other than uv at v. Thus, at least one arc in Π − Σ is B-compatible. Let B f (B c , respectively) be the set of Bfeasible (B-compatible, respectively) arcs in Π− Σ. Note that |Π− Σ| = s − t ≥ k − t − 2 ≥ j ′ − 1. By Lemma 3(2), (4) or (6)  In this case, we construct an (A, A f , A c )-net A and a (B, B f , B c )-net B as in Case 1, except that |A| + |B| = k − 1. We then define one further branch set B 0 that, with A and B, forms the desired K k -minor in X(D).  Recall that t = |Σ ∩ Π|.
Since there is at least one incoming arc at u (because i ≤ k − 3), at least one arc in By Lemma 3(2), (4), (5) or (6), there exists an (A, Similarly, a (B, B f Then B 0 induces a connected subgraph in X(D) by noting that uv is adjacent to both w 1 w ′ 1 and w 2 w ′ 2 . Each branch set of A and B contains an arc outgoing from u or v, which is adjacent to w 1 w ′ 1 or w 2 w ′ 2 . Thus B 0 is adjacent to each branch set of A ∪ B. Hence A ∪ B ∪ {B 0 } forms a K k -minor in X(D). Letā = a be an out-neighbour ofā in F . Note thatā exists sinceāa is not redundant. Then, by the minimality of |Σ ∩ Π|, we haveāā / ∈ Σ ∩ Π. Let B 0 := {uv, au, av,āā}. Then max{|B 0 ∩ Σ|, |B 0 ∩ Π|} ≤ 2 and |B 0 ∩ Σ| + |B 0 ∩ Π| ≤ 3.
Let A := A F (u) − {uv} and B := A F (v). We show that there is a net A at u of size i, and a net B at v of size j, such that A ∪ B ∪ {B 0 } forms a K k -minor in X(D).  (that is, a andā).
In each case, B 0 induces a connected subgraph in X(D). And uv ∈ B 0 is adjacent to each branch set of A, and an arc outgoing from a other than av is adjacent to each branch set of B. Hence A ∪ B ∪ {B 0 } forms a K k -minor in X(D). First suppose that U is not independent in F . That is, there is an arc τ in F joining two vertices in U . Say, τ = a 1 a 2 . Since A F {u, a 2 } is not redundant, in D there is an arc γ = a 2 u outgoing from a 2 . (It may happen that γ ∈ {a 2 a 1 , a 2 v}.) Let B 0 := {uv, τ, γ}. Since uv is adjacent to both τ and γ, B 0 induces a connected subgraph in X(D). Note that max{|B 0 ∩ Σ|, |B 0 ∩ Π|} ≤ 2.
If i ≥ j, then j ≤ k−1 Note that there is at least one (in fact many) incoming arc v l v at v with ϕ(v l , v) / ∈ Σ ∪ B 0 . Thus ϕ(v l , v) ∈ B c and |B c | ≥ 1. By Lemma 3(2), (4) or (6) i and a (B, B f , B c )-net B of size j.
Since each arc outgoing from u or v is adjacent to τ or γ, each branch set of A ∪ B is adjacent to B 0 . Thus, A ∪ B ∪ {B 0 } forms a K k -minor in X(D).
Next suppose that U is independent in F . For each a l ∈ U , if in D there is an arc a l a ′ l other than a l u or a l v, let Q l := {a l a ′ l }. Otherwise, we have A D (a l ) = {a l u, a l v}. Letā l be an in-neighbour other than u, v of a l in F . Then A F (ā l , a l ) =ā l a l . Letā l = a l be an out-neighbour ofā l in F . Let Q l := {a l u, a l v,ā lāl }. Let a l , a m be distinct vertices in U such that w 1 ∈ {a l , a m } when t = 1 and |Q l ∪ Q m | is minimised. Let B 0 := {uv} ∪ Q l ∪ Q m . Note that in X(D) each of the subgraphs induced on Q l and Q m is connected and adjacent to uv, B 0 induces a connected subgraph.
Note that for each p ∈ {l, m}, |Q p ∩ Σ| ≤ 2 and |Q p ∩ Π| ≤ 2. If |Q p ∩ Σ| = 2, then Q p := {a p u, a p v,ā pāp } andā pāp ∈ Σ andā p is adjacent to u (but not v because U is independent) in F . Thusā p a p is A-feasible (A-compatible) ifā pāp is A-feasible (A-compatible). Let Σ ′ be obtained from Σ by replacingā pāp withā p a p . Then |Q p ∩ Σ ′ | ≤ 1 and |B 0 ∩ Σ ′ | ≤ 2. In addition, each element in Σ ′ is A-feasible or A-compatible, and no two share a tail. Similarly, we can obtain Π ′ such that each of its elements is A-feasible or A-compatible, no two elements share a tail and |B 0 ∩ Π ′ | ≤ 2.
Let  Since each arc outgoing from u or v is adjacent to an arc in Q l or Q m , each branch set of A ∪ B is adjacent to B 0 . Thus, A ∪ B ∪ {B 0 } forms a K k -minor in X(D). Since Then each internal vertex of P 0 is not adjacent to u in F . If |V (P 0 ) ∩ N F (v)| = 1, then z m is the only neighbour of v in F which is on P 0 . Let P := P 0 and set z l := z m . If |V (P 0 ) ∩ N F (v)| ≥ 2, let P = (z 1 , z 2 , . . . , z l ) be the subpath of P 0 such that z l ∈ N F (v) and |V (P ) ∩ N F (v)| = 2 .
We shall construct a branch set P ′ consisting of arcs alongside P . Let z 0 = u and z l+1 = v. For 1 ≤ g ≤ l, we associate to z g the set Q g of arcs as follows. If A D (z g ) − (A D {z g−1 , z g } ∪ A D {z g , z g+1 }) = ∅, then let Q g be a singleton set that contains exactly one arc, say, Since the arc A F {z g , z g+1 } ∈ A(F ) is not redundant, z g z g−1 ∈ A D (z g ). Similarly, z g z g+1 ∈ A D (z g ) since A F {z g−1 , z g } ∈ A(F ) is not redundant. Letz g be an in-neighbour of z g in F . Thenz g z g ∈ A(F ). Letz gzg withz g = z g be an arc outgoing fromz g in D (which exists becausē z g z g is not redundant). Set Q g := {z g z g−1 , z g z g+1 ,z gzg }. Note that Q g induces a connected subgraph in X(D) sincez gzg is adjacent to both z g z g−1 and z g z g+1 .
In the case where V (P ) ∩ N F (v) = {z p , z l } (p < l) and Q p = {z p v}, we slightly modify Q p as Let P ′ := ∪ l g=1 Q g . Then, for 1 ≤ g ≤ l − 1, since Q g contains an arc outgoing from z g other than z g z g+1 and Q g+1 contains an arc outgoing from z g+1 other than z g+1 z g , each Q g is adjacent to Q g+1 in X(D). Thus, P ′ induces a connected subgraph in X(D). We call P ′ a parallel set of P .
Let Σ and Π be as above. We have the following claim: There is a set Σ ′ such that |Σ ′ | ≥ |Σ| − 1 and P ′ ∩ Σ ′ = ∅, and each element of which is A-feasible or A-compatible and no two elements share a tail; (b) There is a set Π ′ such that |Π ′ | ≥ |Π| − 2 and P ′ ∩ Π ′ = ∅, and each element of which is B-feasible or B-compatible and no two elements share a tail.
(b) Initially, set Π ′ := Π − P ′ . Recall that P contains at most two neighbours, z g 1 and z g 2 say, of v. Let γ be an arc in Π ∩ P ′ such that there is a Q g containing γ (there may be more than one Q g containing γ) and g / ∈ {g 1 , g 2 }. Since z g is not adjacent to v inH, we have |Q g | = 3 and Q g = {z g z g−1 , z g z g+1 ,z gzg }, wherez g is an in-neighbour of z g in F andz gzg =z g z g is an arc outgoing fromz g in D. Further,z g is a neighbour of v in F and ϕ(v,z g ) =z gzg . Note thatz g z g / ∈ Π is B-feasible or B-compatible. Now update Π ′ by addingz g z g . That is, Π ′ := Π ′ ∪ {z g z g }. By repeating this procedure for all such γ, we obtain a Π ′ with the same size as Π − (Q g 1 ∪ Q g 2 ).
Let B 0 := {uv} ∪ P ′ . Then B 0 induces a connected subgraph in X(D) since uv is adjacent to Q 1 .
Next we show that there exists a net of size i at u and a net of size j at v such that none of their branch sets intersects B 0 .
Since each element of A constructed above contains an arc xx ′ , which is outgoing from a neighbour x = v of u and x ′ = u, each element of A is adjacent to B 0 because uv ∈ B 0 is adjacent to each xx ′ . Note that |V (P ) ∩ N F (v)| ∈ {1, 2}. In the case when |V (P ) ∩ N F (v)| = 1, P ′ contains an arc yy ′ , which is outgoing from an in-neighbour y = u of v and y ′ = v. Since such a yy ′ is adjacent to every arc of A F (v), it is adjacent to every element of B constructed above. In the case when |V (P ) ∩ N F (v)| = 2, P ′ contains two arcs α and β, each of them is outgoing from a neighbour of v other than u and heading to a vertex other than v. Then each arc of A F (v) is adjacent to either α or β. So every element of B is adjacent to P ′ ⊆ B 0 . Therefore, {B 0 } ∪ A ∪ B forms a K k -minor in X(D). Case 2.3. j = k − 2: Then i = 1. Suppose first that d − F (v) = 1; that is, uv is the only incoming arc at v and d F (v) = k − 1. Since v is not special, one out-neighbour v ′ of v in F is not a sink. Now consider the arc vv ′ . If d + F (v ′ ) ≥ 2, then S F (vv ′ ) = d + F (v) + d + F (v ′ ) − 1 ≥ k − 2 + 2 − 1 = k − 1. This is a special case of Case 2.2 and thus can be treated similarly. If d + F (v ′ ) = 1, then by Property A, one potential arc v ′ v ′′ ( = v ′ v) is outgoing from v ′ in D but not present in F (since d + F (v) = 1). Let F ′ be obtained from F by adding v ′ v ′′ . Again we have  This completes the proof of Theorem 1.