THE CHROMATIC DETOUR NUMBER OF A GRAPH

A set $S \subseteq V(G)$ is called a chromatic detour set of $G$ if $S$ is both a chromatic set and a detour set of $G$. The minimum cardinality of a chromatic detour set of $G$ is called a chromatic detour number of $G$ and is denoted by $\chi_{dn}(G)$. Some of its general properties are studied. Connected graphs of order $n \geq 2$ with chromatic detour number $n$ or $n-1$ are characterized. It is shown that for every positive integer $a$ and $b$ with $2 \leq a < b$, there exists a connected graph $G$ such that $dn(G) = a$ and $\chi_{dn}(G) = b$. It is also shown that for every positive integers $a$ and $b$ with $2 \leq a \leq b$, there exists a connected graph $G$ such that $\chi(G) = a$ and $\chi_{dn}(G) = b$.