## Faculty Publications & Research

#### Document Type

Article

#### Publication Date

2008

#### Disciplines

Discrete Mathematics and Combinatorics | Mathematics

#### 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 *K _{s,t} * for a fixed

*s*and large

*t*. Myers proved upper bounds on

*D*(

*K*) and made a conjecture on the order of magnitude of

_{s,t}*D(K*for a fixed

_{s,t})*s*and

*t*→ ∞. He also found exact values for

*D*(

*K*) for an infinite series of

_{2,t}*t*. In this paper, we confirm the conjecture of Myers and find asymptotically (in

*s)*exact bounds on

*D(K*for a fixed

_{s,t})*s*and large

*t*.

#### Recommended Citation

Kostochka, A.V., & Prince, N. (2008). On K_{s,t}-minors in graphs with given average degree. *Discrete Mathematics,* 308(19), 4435-4445.