$[1,1,2]$-Colorings of $K_5$ and $K_6$

Abstract

Given non-negative integers $r,s$, and $t$, an $[r,s,t]$-coloring of a graph $G=(V(G),E(G))$ is a mapping $c$ 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 $|c(v_i)–c(e_i)|\ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$-chromatic number ${\chi }_{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$-coloring. We prove that ${\chi }_{1,1,2}(K_5)=7$ and ${\chi }_{1,1,2}(K_6)=8$.