Clique minors in double‐critical graphs

Abstract

A connected t-chromatic graph G is double-critical if G − {u, v} is (t − 2)-colorable for each edge uv ∈ E(G). A long-standing conjecture of Erdős and Lovász that the complete graphs are the only double-critical t-chromatic graphs remains open for all t ≥ 6. Given the difficulty in settling Erdős and Lovász's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every doublecritical t-chromatic graph contains a K t minor and verified their conjecture for t ≤ 7. Albar and Gonçalves recently proved that every double-critical 8-chromatic graph contains a K 8 minor, and their proof is computer-assisted. In this paper we prove that every double-critical t-chromatic graph contains a K t minor for all t ≤ 9. Our proof for t ≤ 8 is shorter and computer-free.

Erdős–Lovász Tihany Conjecture for graphs with forbidden holes

“…Their proof is computer-assisted. Rolek and the present author [17] gave a computer-free proof of the same result and further showed that any double-critical, t-chromatic graph contains K 9 as a minor for all t ≥ 9. We note here that Theorem 1.3 does not completely settle Conjecture 1.4 for all claw-free graphs.…”

On triangles in ‐minor free graphs

“…Actually, recently Rolek and Song proved Conjecture for , and proved that K 9 ‐minor free graphs are 12‐colorable . Both proofs are obtained without the use of the computer, but Conjectures and remain open.…”

“…This conjecture cannot be extended to the case of 9 -minors. Indeed, for ( 1,2,2,2,2,2 , 6)-cockades (which are 9 -minor free graphs) the ratio between triangles and edges tends to 22 7 > 3 as the number of vertices increases. However, it can be the case that in a 9 -minor free graph the number of 4 satisfies a Mader-like rule with respect to the number of triangles.…”

Graphs with no K9= minor are 10‐colorable

“…To prove Theorem , we will need the following extremal function for minors proved by the present author in .…”

“…We will additionally need the following, which is an abridged version of a key lemma used in the proof of Theorem (see Lemma 2.9 in ). We note that Lemma has been proved by a computer search.…”