Elsevier

Discrete Mathematics

Volume 310, Issue 20, 28 October 2010, Pages 2637-2654
Discrete Mathematics

Dense graphs have K3,t minors

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Abstract

Let K3,t denote the graph obtained from K3,t by adding all edges between the three vertices of degree t in it. We prove that for each t6300 and nt+3, each n-vertex graph G with e(G)>12(t+3)(n2)+1 has a K3,t-minor. The bound is sharp in the sense that for every t, there are infinitely many graphs G with e(G)=12(t+3)(|V(G)|2)+1 that have no K3,t-minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every (s+t)-chromatic graph has a Ks,t-minor.

Keywords

Bipartite minors
Dense graphs

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