Edge mappings of graphs: Turán type parameters

Abstract

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f:E\left(H\right)\to E\left(H\right)$ with $f\left(e\right)\ne e$ for all $e\in E\left(H\right)$ and further in all copies $G^{\prime }$ of $G$ in $H$ there exists $e\in E\left(G^{\prime }\right)$ with $f\left(e\right)\in E\left(G^{\prime }\right)$. Among other results, we determine $h(n,G)$ when $G$ is a matching and $n$ is large enough.

As a related concept, we say that $H$ is unavoidable for $G$ if for any mapping $f:E\left(H\right)\to E\left(H\right)$ with $f\left(e\right)\ne e$ there exists a copy $G^{\prime }$ of $G$ in $H$ such that $f\left(e\right)\notin E\left(G^{\prime }\right)$ for all $e\in E\left(G^{\prime }\right)$. The set of minimal unavoidable graphs for $G$ is denoted by $M\left(G\right)$. We prove that if $F$ is a forest, then $M\left(F\right)$ is finite if and only if $F$ is a matching, and we conjecture that for all non-forest graphs $G$, the set $M\left(G\right)$ is infinite.

Several other parameters are defined with basic results proved. Lots of open problems remain.