Acyclic Coloring of Central and Total Graph of Path \(P_{n}\) and Fan Graph \(F_{m,n}\)

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 \( F_{m,n} \).