Acyclic Coloring of Central and Total Graph of Path $P_n$ and Fan Graph $F_{m,n}$

Abstract

A proper vertex coloring (no two adjacent vertices have the same color) of a graph $G$ is said to be acyclic if the induced subgraph of any two color classes is acyclic. The minimum number of colors required for an acyclic coloring of a graph $G$ is called its acyclic chromatic number and is denoted by $a(G)$. In this paper, we determine the exact value of the acyclic chromatic number for the central and total graphs of the path $P_n$, and the Fan graph $F_{m,n}$.