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A graph $G$ contains $H$ as an \emph{immersion} if there is an injectivemapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$,there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, andall the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogueof Hadwiger's conjecture for the clique immersions by Lescure and Meyniel, andindependently by Abu-Khzam and Langston, states that every graph $G$ contains$K_{\chi(G)}$ as an immersion. We prove that for any constant $\varepsilon>0$and integers $s,t\ge2$, there exists $d_0=d_0(\varepsilon,s,t)$ such that every$K_{s,t}$-free graph $G$ with $d(G)\ge d_0$ contains a clique immersion oforder $(1-\varepsilon)d(G)$. This implies that the above-mentioned conjectureis asymptotically true for graphs without a fixed complete bipartite graph.

Clique immersion in graphs without fixed bipartite graph