Clique immersions in graphs of independence number two with certain forbidden subgraphs

Abstract

The Lescure–Meyniel conjecture is the analogue of Hadwiger’s conjecture for the immersion order. It states that every graph contains the complete graph as an immersion, and like its minor-order counterpart it is open even for graphs with independence number 2. We show that every graph with independence number and no hole of length between 4 and satisfies this conjecture. In particular, every -free graph with satisfies the Lescure–Meyniel conjecture. We give another generalisation of this corollary, as follows. Let and be graphs with independence number at most 2, such that . If is -free, then satisfies the Lescure–Meyniel conjecture.