Forcing a sparse minor

Abstract

This paper addresses the following question for a given graphH: What is the minimum numberf(H) such that every graph with average degree at leastf(H) containsHas a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth whenHis a complete graph. Kostochka and Thomason independently proved that$f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determinedf(H) whenHhas a super-linear number of edges. We focus on the case whenHhas a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that ifHhastvertices and average degreedat least some absolute constant, then$f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case whenHhas small average degree, we prove that ifHhastvertices andqedges, thenf(H) ⩽t+ 6.291q(where the coefficient of 1 in thetterm is best possible).