Derangement Representation of Graphs

Abstract

A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v$ and $u$ are adjacent if and only if $\pi (v)(i)\ne \pi (u)(i)$ for each $i\in \{1,2,3,\dots ,k\}$. The derangement representation number of $G$ denoted by $drn(G)$, is the minimum of $k$ such that $G$ has a derangement $k$-representation. In this paper, we prove that any graph has a derangement $k$-representation. Also, we obtain some lower and upper bounds for $drn(G)$, in terms of the basic parameters of $G$. Finally, we determine the exact value or give the better bounds of the derangement representation number of some classes of graphs.

20 pages, 3 figures, 4 tables