Average degree conditions forcing a minor (Q259170)

Summary: Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph \(H\) as a minor. Here, we strengthen (under certain conditions) a recent result by \textit{B. A. Reed} and \textit{D. R. Wood} [``Forcing a sparse minor'', Comb. Probab. Comput. 25, No. 2, 300--322 (2016; \url{doi:10.1017/S0963548315000073}], giving better bounds on the average degree required to force an \(H\)-minor when \(H\) is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when \(H\) is an unbalanced complete bipartite graph.