The extremal function for K9= minors

Abstract

We prove the extremal function for K9= minors, where K9= denotes the complete graph K9 with two edges removed. In particular, we show that any graph with n≥8 vertices and at least 6n−20 edges either contains a K9= minor or is isomorphic to a graph obtained from disjoint copies of K8 and K2,2,2,2,2 by identifying cliques of size 5. We utilize computer assistance to prove one of our lemmas.

Graphs with no K9= minor are 10‐colorable

“…It is difficult to improve Theorem 1.7 for small values of k. Recently, Liu, the present author, and Yu [10,11] have been able to show that for ∈ k m ( , ) { (15,8), (27,9), (43, 10)} 0 , and ≥ k k 0 , any k-contraction-critical graph is m-connected. Some improvements for larger k have also been found, for example [8,9].…”

“…Theorem 1. 4 Rolek [15]. If G is a graph with ≥ G | | 8 and at least G 6 | | − 20 edges, then either ≥ -cockade is a graph built up from disjoint copies of K 8 and K 2,2,2,2,2 by identifying cliques of size 5.…”

“…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.…”

Graphs with no $K_9^=$ minor are 10-colorable

“…To prove Theorem 1.3, we will need the following extremal function for K = 9 minors proved by the present author in [11].…”