$[r,s,t]$-Colorings of Friendship Graphs and Wheels

Abstract

Given non-negative integers $r,s$ and $t$, an $[r,s,t]$-coloring of a graph $G=(V,E)$ is a mapping $c$ from $V\cup E$ to the color set $\{0,1,\ldots ,k-1\}$ such that $|c(v_{i})-c(v_{j})|\ge r$ for every two adjacent vertices $v_{i}$, $v_{j}$, $|c(e_{i})-c(e_{j})|\ge s$ for every two adjacent edges $e_{i}$, $e_{j}$, and $|c(v_{i})-c(e_{j})|\ge t$ for every vertex $v_{i}$ and an incident edge $e_{j}$, respectively. The minimum $k$ such that $G$ admits an $[r,s,t]$-coloring is called the $[r,s,t]$-chromatic number of $G$ and is denoted by $\chi _{r,s,t}(G)$. In this paper, we examine exact values and upper bounds for the $[r,s,t]$-chromatic number of friendship graphs and wheels for every positive integer $r,s$ and $t$.