Fractional f-Edge Cover Chromatic Index of Graphs

Abstract

Let G be a graph, and let f be an integer function on V with ${1\leq f(v)\leq d(v)}$ to each vertex ${\upsilon \in V}$. An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex ${\upsilon \in V(G)}$ at least f(υ) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by ${\chi^{'}_{fc}(G)}$. It is well known that any simple graph G has the f-edge cover chromatic index equal to δ _f (G) or δ _f (G) − 1, where ${\delta_{f}(G)=\,min\{\lfloor \frac{d(v)}{f(v)} \rfloor: v\in V(G)\}}$. The fractional f-edge cover chromatic index of a graph G, denoted by ${\chi^{'}_{fcf}(G)}$, is the fractional f-matching number of the edge f-edge cover hypergraph ${\mathcal{H}}$ of G whose vertices are the edges of G and whose hyperedges are the f-edge covers of G. In this paper, we give an exact formula of ${\chi^{'}_{fcf}(G)}$ for any graph G, that is, ${\chi^{'}_{fcf}(G)=\,min \{\min\limits_{v\in V(G)}d_{f}(v), \lambda_{f}(G)\}}$, where ${\lambda_{f}(G)=\min\limits_{S} \frac{|C[S]|}{\lceil (\sum\limits_{v\in S}{f(v)})/2 \rceil}}$ and the minimum is taken over all nonempty subsets S of V(G) and C[S] is the set of edges that have at least one end in S.