On the \(f\)-Chromatic Class of a Multi-Wheel Graph

Abstract

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. A multi-wheel graph is a graph obtained from \(s\) cycles \(C_{n_1}, C_{n_2}, \ldots, C_{n_s}\) (\(s \geq 1\)) by adding a new vertex, say \(w\), and edges joining \(w\) to all the vertices of the \(s\) cycles. In this article, we solve a conjecture posed by Yu et al. in 2006 and prove that it is not always true. Furthermore, the classification problem of multi-wheel graphs on \(f\)-colorings is solved completely.