Clique Immersions In Graphs Of Independence Number Two With Certain Forbidden Subgraphs

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 Kχ(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 α(G)≥2 and no hole of length between 4 and 2α(G) satisfies this conjecture. In particular, every C4-free graph G with α(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)|≤4. If G is H-free, then G satisfies the Lescure–Meyniel conjecture.