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Register a new userThe ErdH{o}s-Hajnal property for graphs with no fixed cycle as a pivot-minor
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We prove that for every integer $k$, there exists $varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B subseteq V(G)$ such that $|A|,|B| geq varepsilon n$, and $A$ is either complete or anticomplete to $B$. This proves the analog of the ErdH{o}s-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to $C_k$.
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Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the ErdH{o}s-Posa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the ErdH{o}s-Posa property, and indeed even fails when $H$ is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.
We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. This answers a question of Kim and Kwon [ErdH{o}s-Posa property of chordless cycles and its applications. JCTB 2020].
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A hole in a graph is an induced cycle of length at least $4$. Let $sge2$ and $tge2$ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $chi(G[S ]) ge s$ and $chi(G[T ]) ge t$. The well-known ErdH{o}s-Lovasz Tihany Conjecture from 1968 states that every graph $G$ with $omega(G) < chi(G) = s + t - 1$ is $(s,t)$-splittable. This conjecture is hard, and few related results are known. However, it has been verified to be true for line graphs, quasi-line graphs, and graphs with independence number $2$. In this paper, we establish more evidence for the ErdH{o}s-Lovasz Tihany Conjecture by showing that every graph $G$ with $alpha(G)ge3$, $omega(G) < chi(G) = s + t - 1$, and no hole of length between $4$ and $2alpha(G)-1$ is $(s,t)$-splittable, where $alpha(G)$ denotes the independence number of a graph $G$.
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