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(k) independent of G such that the complement of G has a minor isomorphic to Ks, where s = d2 (1 + 1/t)n − ce. We prove that Kostochka’s 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 Kostochka’s conjecture holds for all integers t ≤ 5, and that a weaker form with s replaced by s′ = d2(1 + 1/(2t))n − ce holds for all integers t ≥ 1. 12 May 1998, revised 2 May 1999. Published in J. Combin. Theory Ser. B 78 (2000), 81–85. * Collaboration supported by a CNRS-NSF international cooperation research grant. ** Supported in part by NSF under Grant No. DMS-9303761, by ONR under Contract number N0001493-1-0325, and by ProNEx (MCT/FINEP) (Proj. 107/97) and FAPESP (Proc. 97/14469–6 and Proc. 96/04505–2).