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 $C_{n_1},C_{n_2},\dots ,C_{n_s}$ ($s\ge 1$) 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.