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Conjecture 1.2 (The Erd˝os-Lov´asz Tihany conjecture). Let G be a graph with ω(G) < χ(G) = s + t − 1, where t ≥ s ≥ 2 are integers. Then G is (s,t)-splittable.

Conjecture 1.2 is hard, and few related results are known. The case (2, 2) for Conjecture 1.2 is trivial; the cases (2, 3) and (3, 3) were shown by Brown and Jung [BJ69] in 1969; Mozhan [Moz87] and Stiebitz [Sti87] each independently showed the case (2, 4) in 1987; the cases (3, 4) and (3, 5) were settled by Stiebitz [Sti88] in 1988. A relaxed version of Conjecture 1.2 was proved in [Sti17]. Recent work on both Conjecture 1.1 and Conjecture 1.2 have also focused on proving the conjectures for certain classes of graphs. A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. Note that every is quasi-line and every quasi-line graph is claw-free [CS12]. A hole in a graph is an induced of length at least four; a hole is even if it has an even length. A graph is even-hole-free if it contains no even hole. Hadwiger’s conjecture has been shown to be true for line graphs by Reed and Seymour [RS04]; quasi-line graphs by Chudnovsky and Ovetsky Fradkin [CF08]; graphs G with α(G) ≥ 3 and no hole of length between 4 and 2α(G)−1 by Thomas and the present author [TS17]. Meanwhile, the Erd˝os-Lov´asz Tihany conjecture has also been verified to be true for line graphs by Kostochka and Stiebitz [KS08]; quasi-line graphs, and graphs G with α(G) = 2 by Balogh, Kostochka, Prince and Stiebitz [BKPS09]; graphs G with α(G) ≥ 3 and no hole of length between 4 and 2α(G) − 1 by the present author [Son19]. Chudnovsky and Seymour [CS19] recently proved a structural result on even-hole-free graphs.

Theorem 1.3 (Chudnovsky and Seymour [CS19]). Let G be a non-empty even-hole-free graph. Then G has a bisimplicial vertex and χ(G) ≤ 2ω(G) − 1.

It is unknown whether Conjecture 1.1 and Conjecture 1.2 hold for even-hole-free graphs. Using

Theorem 1.3, we prove in Section 2 that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 6)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies Conjecture 1.2 provided that t ≥ s>χ(G)/3. It is worth noting that Kawarabayashi, Pedersen and Toft [KPT11] observed that if Hadwiger’s conjecture holds, then the following conjecture might be easier to settle than the Erd˝os-Lov´asz Tihany conjecture.

Conjecture 1.4 (Kawarabayashi, Pedersen, Toft [KPT11]). Every graph G satisfying ω(G) <

χ(G) = s + t − 1 has two vertex-disjoint subgraphs G1 and G2 such that G1 < Ks and G2 < Kt, where t ≥ s ≥ 2 are integers.

2 In the same paper [KPT11], they settled Conjecture 1.4 for a few additional values of (s,t) ∈ {(2, 6), (3, 6), (4, 4), (4, 5)}. We end Section 2 by proving the (4, 6) case for Conjecture 1.4, that is, we prove that every graph G with χ(G) = 9 >ω(G) has a K4 ∪ K6 minor. Here K4 ∪ K6 denotes the disjoint union of K4 and K6.

We need to introduce more notation. Let G be a graph. For a vertex x ∈ V (G), we will use N(x) to denote the set of vertices in G which are adjacent to x. We define N[x]= N(x) ∪ {x} and d(x) = |N(x)|. If A, B ⊆ V (G) are disjoint, we say that A is complete to B if each vertex in A is adjacent to all vertices in B, and A is anti-complete to B if no vertex in A is adjacent to any vertex in B. If A = {a}, we simply say a is complete to B or a is anti-complete to B. The subgraph of G induced by A, denoted G[A], is the graph with vertex set A and edge set {xy ∈ E(G) : x,y ∈ A}. We denote by B \ A the set B − A, and G \ A the subgraph of G induced on V (G) \ A, respectively. If A = {a}, we simply write B \a and G\a, respectively. We say that G is k-chromatic if χ(G)= k. An (s,t)-graph is a connected (s+t−1)-chromatic graph which does not contain two vertex-disjoint subgraphs with chromatic number s and t, respectively. We use the convention “A :=” to mean that A is defined to be the right-hand side of the relation.

Finally, we shall make use of the following results of Stiebitz [Sti88, Sti96] and Mader [Mad68].

Theorem 1.5 (Stiebitz [Sti88]). Suppose G is an (s,t)-graph with t ≥ s ≥ 2. If ω(G) ≥ t, then ω(G) ≥ s + t − 1.

Theorem 1.6 (Stiebitz [Sti96]). Every graph G satisfying δ(G) ≥ s + t + 1 has two vertex-disjoint subgraphs G1 and G2 such that δ(G1) ≥ s and δ(G2) ≥ t.

Theorem 1.7 (Mader [Mad68]). For every integer p ≤ 7, every graph on n ≥ p vertices and at p−1 least (p − 2)n − 2  + 1 edges has a Kp minor.

2 Main results

Rolek and the present author [RS17, Theorem 5.2] proved that if Mader’s bound in Theorem 1.7 can be generalized to all values of p (as in [RS17, Conjecture 5.1]), then every graph with no Kp minor is (2p − 6)-colorable for all p ≥ 7. It is hard to prove [RS17, Conjecture 5.1]. We begin this section with an easy result on coloring even-hole-free graphs with no Kk minor, where k ≥ 7. It seems non-trivial to improve the bound in Theorem 2.1 to 2k − 6.

Theorem 2.1. For all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable.

Proof. Suppose the assertion is false. Let G be an even-hole free graph with no Kk minor and χ(G) ≥ 2k − 4. We choose G with |G| minimum. Then G is vertex-critical and χ(G) = 2k − 4. Thus δ(G) ≥ 2k − 5; in addition, G is connected and has no clique-cut. Suppose ω(G) ≥ k − 1. Let K be a (k − 1)-clique in G. Then G \ K is connected because G has no clique-cut; by contracting

G \ K into a single vertex we obtain a Kk minor, a contradiction. Thus ω(G) ≤ k − 2. Since G is even-hole-free, by Theorem 1.3, χ(G) ≤ 2ω(G) − 1 ≤ 2(k − 2) − 1 = 2k − 5, a contradiction.

3 We next prove a lemma which plays a key role in the proof of Theorem 2.3 and Theorem 2.4.

Lemma 2.2. Let G be a graph and x ∈ V (G) with p := χ(G[N(x)]) ≥ 2. Let V1,...,Vp be the color classes of a proper p-coloring of G[N(x)] with |V1|≥···≥|Vp|≥ 1. If |Vr ∪···∪ Vp|≤ χ(G) − r − 1 for some r ∈ [p] with 2 ≤ r ≤ p, then p ≤ χ(G) − 2 and G is (r, χ(G) + 1 − r)-splittable.

Proof. Let G, p, r, V1,...,Vp be as given in the statement. Note that p − r + 1 ≤ |Vr ∪···∪ Vp|≤

χ(G) − r − 1 and so p ≤ χ(G) − 2 and V (G) \ N[x] 6= ∅. Let W := V1 ∪···∪ Vr−1. Then χ(G[{x}∪ W ]) = r and χ(G \ W ) ≥ χ(G) − (r − 1) = χ(G) + 1 − r. It suffices to show that χ(G \ ({x}∪ W )) ≥ χ(G \ W ). Let q := χ(G \ ({x}∪ W )) ≥ χ(G \ W ) − 1 ≥ χ(G) − r ≥ 2 and let U1,...,Uq be the color classes of a proper q-coloring of G \ ({x}∪ W ). Since x is adjacent to

|Vr ∪···∪ Vp|≤ χ(G) − r − 1 ≤ q − 1 vertices in G \ W , we see that x is anti-complete to Ui for some i ∈ [q]. We may assume that i = 1. Then U1 ∪ {x},U2,...,Uq form the color classes of a proper q-coloring of G \ W . Therefore, χ(G \ ({x}∪ W )) = q ≥ χ(G \ W ) ≥ χ(G) − r + 1, as desired.

We are now ready to prove that the Erd˝os-Lov´asz Tihany conjecture holds for even-hole-free graphs G with ω(G) < χ(G) = s + t − 1 if t ≥ s>χ(G)/3. It would be nice if one can prove the same holds without the additional condition.

Theorem 2.3. Let G be an even-hole-free graph with ω(G) < χ(G)= s + t − 1, where t ≥ s ≥ 2. If s>χ(G)/3, then G is (s,t)-splittable.

Proof. Suppose the assertion is false. Let G be a counterexample with |G| minimum. Then G is vertex-critical; in addition, G is an (s,t)-graph. Thus δ(G) ≥ χ(G)−1= s+t−2. By Theorem 1.5, ω(G) ≤ t − 1. Since G is even-hole-free, by Theorem 1.3, G has a bisimplicial vertex v such that N(v) is the union of two cliques. Thus α(G[N(v)]) ≤ 2, ω(G[N(v)]) ≤ t − 2 and

s + t − 2= χ(G) − 1 ≤ δ(G) ≤ d(v) ≤ 2ω(G[N(v)]) ≤ 2t − 4.

It follows that t ≥ s + 2 ≥ 4 and χ(G)= s + t − 1 ≥ 2s + 1. We next claim that ∆(G) ≤ |G|− 2. Suppose there exists x ∈ V (G) such that d(x)= |G|− 1. Then

χ(G \ x)= χ(G) − 1= s + (t − 1) − 1 >ω(G) − 1= ω(G \ x) and t − 1 >s>χ(G \ x)/3.

By the minimality of G, G \ x is (s,t − 1)-splittable and thus G is (s,t)-splittable, a contradiction. Thus ∆(G) ≤ |G|− 2, as claimed. It follows that V (G) \ N[v] 6= ∅ and so χ(G[N[v]]) ≤ χ(G) − 1. Let p := χ(N(v)). Then p = χ(G[N[v]]) − 1 ≤ χ(G) − 2. Note that

p ≥ ω(G[N(v)]) ≥ d(v)/2 ≥ (χ(G) − 1)/2 ≥ ((2s + 1) − 1)/2= s ≥ 2.

Let V1,...,Vp be the color classes of a proper p-coloring of G[N(v)] with 2 ≥ |V1|≥···≥|Vp|≥ 1.

Suppose p ≥ t − 1. Then |Vt−2| = 1 because d(v) ≤ 2t − 4. Therefore,

|Vt ∪···∪ Vp| = p − t + 1 ≤ (χ(G) − 2) − t +1= χ(G) − t − 1.

By Lemma 2.2 applied to G and v with r = t, we see that G is (s,t)-splittable, a contradiction.

Thus s ≤ p ≤ t − 2. Next, if |Vs ∪···∪ Vp|≤ χ(G) − s − 1, then G is (s,t)-splittable by applying

4 Lemma 2.2 to G and v with r = s, a contradiction. Hence, |Vs ∪···∪ Vp|≥ χ(G) − s = t − 1 ≥ 3.

Note that p − s + 1 ≤ (t − 2) − 2+1= t − 3, and so |Vs| = 2 and

d(v) = (|V1| + · · · + |Vs−1|)+ |Vs ∪···∪ Vp|≥ 2(s − 1) + t − 1 = 2s + t − 3.

It follows that t − 2 ≥ ω(G[N(v)]) ≥ d(v)/2 ≥ (2s + t − 3)/2, which implies that t ≥ 2s +1. Thus χ(G)= s + t − 1 ≥ 3s, contrary to the assumption that 3s>χ(G).

Finally, we prove that Conjecture 1.4 is true when (s,t) = (4, 6).

Theorem 2.4. Every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor.

Proof. Suppose for a contradiction that G is a counterexample to the statement with minimum number of vertices. Then G is vertex-critical, and so δ(G) ≥ 8 and G is connected. Suppose G contains two vertex-disjoint subgraphs G1 and G2 such that χ(G1) ≥ 4 and χ(G2) ≥ 6. Since Hadwiger’s conjecture holds for k-chromatic graphs with k ≤ 6, we see that G1 < K4 and G2 < K6, a contradiction. Thus G is a (4, 6)-graph, and so ω(G) ≤ 5 by Theorem 1.5. Note that G is not necessarily contraction-critical, as a proper minor of G may have clique number 9. We claim that

Claim 1. 2 ≤ α(G[N(x)]) ≤ d(x) − 7 for each x ∈ V (G).

Proof. Let x ∈ V (G). Since ω(G) ≤ 5 and δ(G) ≥ 8, we see that α(G[N(x)]) ≥ 2. Suppose α(G[N(x)]) ≥ d(x) − 6. Let A be a maximum independent set of G[N(x)]. Let G∗ be obtained from G by contracting G[A ∪ {x}] into a single vertex, say w. Note that ω(G∗) < 8 and G∗ has no ∗ ∗ K4 ∪ K6 minor. By the minimality of G, χ(G ) ≤ 8. Let c : V (G ) → [8] be a proper 8-coloring of G∗. Since |N(x) \ A| = d(x) − |A|≤ 6, we may assume that c(N(x) \ A) ⊆ [6] and c(w) = 7. But then we obtain a proper 8-coloring of G from c by coloring all the vertices in A with color 7 and the vertex x with color 8, a contradiction. Thus 2 ≤ α(G[N(x)]) ≤ d(x) − 7, as claimed.

By Claim 1, δ(G) ≥ 9. Suppose δ(G) ≥ 13. By Theorem 1.6, G contains two vertex-disjoint subgraphs G1 and G2 such that δ(G1) ≥ 4 and δ(G2) ≥ 8. By Theorem 1.7, we see that G1 < K4 and G2 < K6, a contradiction. Thus 9 ≤ δ(G) ≤ 12. We next claim that

Claim 2. G[N(x)] is even-hole-free and χ(G[N(x)]) ≤ 2ω(G[N(x)]) − 1 for each x ∈ V (G).

Proof. Let x ∈ V (G). Suppose G[N(x)] contains an even hole C. Then χ(G[V (C) ∪ {x}]) = 3 and so χ(G \ (V (C) ∪ {x})) ≥ χ(G) − 3 = 6. It is easy to see that G[V (C) ∪ {x}] < K4. Since Hadwiger’s conjecture holds for 6-chromatic graphs, we see that G \ (V (C) ∪ {x}) has a K6 minor, and so G has a K4 ∪ K6 minor, a contradiction. Thus G[N(x)] is even-hole-free. By Theorem 1.3, χ(G[N(x)]) ≤ 2ω(G[N(x)]) − 1.

Let v ∈ V (G) with d(v) = δ(G), and let p := χ(G[N(v)]). Since 9 ≤ d(v) ≤ 12, we see that p ≥ 3 by Claim 1. Suppose G[N(v)] is K3-free. By Claim 2, p ≤ 2ω(G[N(v)]) − 1 = 3. Thus χ(G[N[v]]) = 4 and χ(G \ N[v]) = χ(G \ N(v)) ≥ 9 − 3 = 6, contrary to the fact that G is

5 a (4, 6)-graph. Thus ω(G[N(v)]) ≥ 3. Let v1, v2, v3 ∈ N(v) be pairwise adjacent in G and let H := G \ {v, v1, v2, v3}. Then G[{v, v1, v2, v3}]= K4 and

2e(H) ≥ (d(v) − 3)(|G \ N[v]|) + (d(v) − 4) · |N(v) \ {v1, v2, v3}| = (d(v) − 3)(|G|− d(v) − 1) + (d(v) − 4)(d(v) − 3) = (d(v) − 3)(|H|− 1).

Suppose d(v) ∈ {11, 12}. Then 2e(H) ≥ 8(|H|− 1). By Theorem 1.7, H < K6, and so G has a K4 ∪ K6 minor, a contradiction. This proves that 9 ≤ d(v) ≤ 10. Then p ≥ 4 by Claim 1. Since ω(G[N(v)]) ≤ 4, we see that G[N(v)] has an anti-matching of size at least three. It follows that

4 ≤ p ≤ d(v) − 3. Let V1,...,Vp be the color classes of a proper p-coloring of G[N(v)] with |V1|≥

···≥|Vp|≥ 1. If p ∈ {4, 5}, then |V4|≤ 2 because d(v) ≤ 10. Thus |V4 ∪···∪Vp|≤ 4= χ(G)−4−1. By Lemma 2.2 applied to G and v with r = 4, we see that G is (4, 6)-splittable, contrary to the fact that G is a (4, 6)-graph. It remains to consider the case 6 ≤ p ≤ d(v) − 3. Since d(v) ≤ 10, we see that |V5| = 1. Thus |V6 ∪···∪ Vp| = p − 5 ≤ (d(v) − 3) − 5 ≤ 2= χ(G) − 6 − 1. By Lemma 2.2 applied to G and v with r = 6, we see that G is (4, 6)-splittable, a contradiction.

This completes the proof of Theorem 2.4.

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