Derangement Representation of Graphs

TL;DR

This paper introduces the concept of derangement $k$-representations of graphs, proves their existence for all graphs, and provides bounds and exact values for specific graph classes.

Contribution

It defines the derangement representation number of graphs, proves its universal existence, and derives bounds and exact values for certain graph classes.

Findings

Every graph admits a derangement $k$-representation.

Bounds for $drn(G)$ are established based on graph parameters.

Exact values or improved bounds are found for specific graph classes.

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 $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in \{1,2,3,\ldots,k\}$. 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.