Elsevier

Discrete Mathematics

Volume 308, Issue 19, 6 October 2008, Pages 4435-4445
Discrete Mathematics

On Ks,t-minors in graphs with given average degree

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Abstract

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for ‘balanced’ H random graphs provide extremal examples and determine the extremal function. Examples of ‘unbalanced graphs’ are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t) and made a conjecture on the order of magnitude of D(Ks,t) for a fixed s and t. He also found exact values for D(K2,t) for an infinite series of t. In this paper, we confirm the conjecture of Myers and find asymptotically (in s) exact bounds on D(Ks,t) for a fixed s and large t.

Keywords

Graph minors
Average degree
Complete bipartite graphs

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This material is based upon work partially supported by the National Science Foundation under Grants DMS-0099608 and DMS-0400498 and by Grants 05-01-00816 and 06-01-00694 of the Russian Foundation for Basic Research.