Contractions to k₈

Abstract

It is proved that the maximal number of edges in a graph with n L 8 vertices that is not contractible to & is 6n -21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n -21 edges that are not contractible to KS are the K5(2)-cockades that have exactly 6n -20 edges. 0 1994 John Wiley & Sons, Inc.is a Ks(2)-cockade.If a graph G is contractible to a complete graph K,,, then one of the simplicia1 summands is contractible to K,,. Therefore Ks(2)-cockades are not By the above definition, a K5(2)-cockade, (MPi-cockade) is the union of some spanned subgraphs that are isomorphic to K5(2) (graphs obtained by adding i vertices to a K4 or a 4-connected maximal planar graph). These unique subgraphs are called the cockade-members.A cockade-member of a Kg(2)-cockade (MPi-cockade) is an end-member if it has at most one K5 (K3+i) that is also contained in another cockademember. CONTRACTIONS TO Ka 433The minimum degree and the maximum degree of a graph G is denoted Otherwise we use standard notation and terminology, see Bollob6s [3].

Minor extremal problems using Turan graphs

“…We have gotten a new lower bound that approaches the exact solution of the problem for infinitely many related values of n and p. Indeed, we think that if p ≤ n ≤ g(p), with g(p) ≈ 2p − 6, it seems impossible to find out a lower bound that improves the new one obtained in terms of Turán Numbers. Furthermore this bound is optimum for 6 ≤ p ≤ 8 (see [7] and [12]) and if 5n+9 8 ≤ p ≤ n (see [2,3]). Hence, an interesting open problem would be to prove the following conjecture.…”

“…But it only suffices to see the exact value ex(n; MK 8 ) = 6n − 20, if 5 divides n 6n − 21, otherwise obtained by Jorgensen [7] to check that (1) is not optimum yet. In this paper we find a new lower bound for the extremal number ex(n; MK p ) that improves (1) and it is best possible for infinitely many values of n and p. This lower bound also allows us to prove that every Turán Graph T r (n) contains K p as a minor for all n ≥ 2p − 2.…”

K6 minors in large 6-connected graphs

“…(We use \ for deletion.) Jørgensen [4] made the following beautiful conjecture. Conjecture 1.1 Every 6-connected graph with no K 6 minor is apex.…”

Clique minors in double‐critical graphs

“…Jørgensen [5] and later Song and Thomas [9] generalized Theorem 1.8 to p = 8 and p = 9, respectively, as follows.…”

“…Theorem 1.9 (Jørgensen [5]) Every graph on n ≥ 8 vertices with at least 6n − 20 edges either contains a K 8 -minor or is isomorphic to a (K 2,2,2,2,2 , 5)-cockade. Theorem 1.10 (Song and Thomas [9]) Every graph on n ≥ 9 vertices with at least 7n − 27 edges either contains a K 9 -minor, or is isomorphic to K 2,2,2,3,3 , or is isomorphic to a (K 1,2,2,2,2,2 , 6)-cockade.…”