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

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 onD(H) for almost all graphsH 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. © 2007 Elsevier B.V. All rights reserved.