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

The Lescure–Meyniel conjecture is the analogue of Hadwiger’s for immersion order. It states that every graph G contains complete Kχ(G) as an immersion, and like its minor-order counterpart it open even graphs with independence number 2. We show α(G)≥2 no hole length between 4 2α(G) satisfies this conjecture. In particular, C4-free α(G)=2 give another generalisation corollary, follows. Let H be at most 2, such |V(H)|≤4. If H-free, then