$[r,s,t]$-colorings of Fans

Abstract

Given non-negative integers $r$, $s$, and $t$, an $[r,s,t]-coloring$ of a graph $G=(V(G),E(G))$ is a function $c$ from $V(G)\cup E(G)$ to the color set $\{0,1,\dots ,k-1\}$ 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 all pairs of incident vertices $v_i$ and edges $e_j$. The [$r$, $s$, $t$]-chromatic number ${\chi }_{r,s,t}(G)$ is the minimum $k$ such that $G$ admits an [$r$, $s$, $t$]-coloring. In this paper, we examine [$r$, $s$, $t$]-chromatic numbers of fans for every positive integer $r$, $s$, and $t$.