Strong Local Colorings of Coronas

Abstract

In this paper, we study strong local colorings of some important families of coronas. A local coloring of a graph G of order at least 2 is a function $c : V (G) \to N$ such that for every set $S\subseteq V (G)$ with $2\le |S| \le 3,$ there exists two distinct vertices $u,v\in S$ such that $|c(u)-c(v)|\ge m_s,$ where $m_s$ is the size of the induced subgraph . The value of a local coloring c is the maximum color it assigns to a vertex of G. The local chromatic number of $G$ is the minimum value of any local coloring of G and we denote it by $\chi_\ell(G).$ A local coloring of $G$ with value $\chi_{\ell}(G)$ is called a minimum local coloring of G. If a minimum local coloring of G uses all the $\chi_\ell(G)$ colors then it is called a strong local coloring of G. If every minimum local coloring of G uses all the $\chi_\ell(G)$ colors then G is called strong local colorable and in this case, its local chromatic number is called strong local chromatic number and is denoted by $\chi_{s\ell}(G).$ In this paper, we have considered some important families of coronas and determined the strong local chromatic number, if it exists; otherwise, we have proved that they are not strong local colorable but local colorable and determined their local chromatic number. Keywords: Local coloring; strong local coloring; local chromatic number; strong local chromatic number; strong local colorable.