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arXiv · 2011.10961

Clique immersion in graphs without fixed bipartite graph

Hong Liu·Guanghui Wang·Donglei Yang
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Abstract

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, 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 $K_{\chi(G)}$ 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 of order $(1-\varepsilon)d(G)$. This implies that the above-mentioned conjecture is asymptotically true for graphs without a fixed complete bipartite graph.

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Hong Liu, Guanghui Wang, Donglei Yang. 2020-11-22. Clique immersion in graphs without fixed bipartite graph. https://arxiv.org/abs/2011.10961

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