On Seymour’s strengthening of Hadwiger’s conjecture for graphs with certain forbidden subgraphs

Let H be a set of graphs. A graph is called H-free if it does not contain a copy of a member of H as an induced subgraph. If H is a graph then G is called H-free if it is {H}-free. Plummer, Stiebitz, and Toft proved that, for every -free graph H on at most four vertices, every -free graph G has a collection of ⌈|V(G)|/2⌉ many pairwise adjacent vertices and edges (where a vertexvand an edgeeare adjacent if v is disjoint from the set V(e) of endvertices of e and adjacent to some vertex of V(e), and two edgeseandfare adjacent if V(e) and V(f) are disjoint and some vertex of V(e) is adjacent to some vertex of V(f)). Here we generalize this statement to -free graphs H on at most five vertices.