Limits of degeneracy for colouring graphs with forbidden minors

Motivated by Hadwiger’s conjecture, Seymour asked which graphs H H have the property that every non-null graph G G with no H H minor has a vertex of degree at most | V ( H ) | − 2 |V(H)|-2 . We show that for every monotone graph family F F with strongly sublinear separators, all sufficiently large bipartite graphs H ∈ F H ∈ F with bounded maximum degree have this property. None of the conditions that H H belongs to F F , that H H is bipartite and that H H has bounded maximum degree can be omitted.