\([r, s, t]\)-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 \(\{0, 1, \ldots, k-1\}\) such that \(|c(v_i) – c(v_j)| \geq r\) for every two adjacent vertices \(v_i\), \(v_j\), \(|c(e_i) – c(e_j)| \geq s\) for every two adjacent edges \(e_i\), \(e_j\), and \(|c(v_i) – c(e_j)| \geq t\) 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 \(t\).