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

Let D(H)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)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,tKs,t for a fixed s and large t  . Myers proved upper bounds on D(Ks,t)D(Ks,t) and made a conjecture on the order of magnitude of D(Ks,t)D(Ks,t) for a fixed s   and t→∞t→∞. He also found exact values for D(K2,t)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)D(Ks,t) for a fixed s and large t.