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 $G$ contains the complete graph $K_{\chi(G)}$ as an immersion, and like its minor-order counterpart it is open even for graphs with independence number 2. We show that every graph $G$ with independence number $\alpha(G)\ge 2$ and no hole of length between $4$ and $2\alpha(G)$ satisfies this conjecture. In particular, every $C_4$-free graph $G$ with $\alpha(G)= 2$ satisfies the Lescure-Meyniel conjecture. We give another generalisation of this corollary, as follows. Let $G$ and $H$ be graphs with independence number at most 2, such that $|V(H)|\le 4$. If $G$ is $H$-free, then $G$ satisfies the Lescure-Meyniel conjecture.Comment: 14 pages, 3 figures. The statements of lemmas 3.1, 4.1, and 4.2 are slightly changed from the previous version in order to fix some minor errors in the proofs of theorems 3.2 and 4.3. Shorter proof of Proposition 5.2 give