Abstract
A graph G contains H as an immersion if there is an injective mapping ϕ:V(H)→V(G) such that for each edge uv∈E(H), there is a path Puv in G joining vertices ϕ(u) and ϕ(v), and all the paths Puv, uv∈E(H), are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel, and independently by Abu-Khzam and Langston, states that every graph G contains Kχ(G) as an immersion. We prove that for any constant ε>0 and integers s,t≥2, there exists d0=d0(ε,s,t) such that every Ks,t-free graph G with d(G)≥d0 contains a clique immersion of order (1−ε)d(G). This implies that the above-mentioned conjecture is asymptotically true for graphs without a fixed complete bipartite graph.
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