Clique Minors in Graphs and Their Complements

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t⩾1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c=c(t) independent of G such that the complement of G has a minor isomorphic to Ks, where s=⌈12(1+1/t) n−c⌉. We prove that Kostochkaʹs conjecture is equivalent to the conjecture of Duchet and Meyniel that every graph with no minor isomorphic to Kt+1 has an independent set of size at least n/t. We deduce that Kostochkaʹs conjecture holds for all integers t⩽5, and that a weaker form with s replaced by s′=⌈12(1+1/(2t)) n−c⌉ holds for all integers t⩾1.