Abstract:

In this paper all graphs are simple and finite, so they
will have no loops or multiple edges. A graph $G$ with vertex set
$V$ and edge set $E$ is denoted by $G=(V,E)$. In an ordinary
edge cover-colouring, each colour appears at every vertex at
least one time. The maximum number of colours needed for an edge
cover colouring of $G$ is called the edge cover chromatic
index of $G$, denoted by $\chi^{'}_{c}(G)$. Gupta's theorem$^{[1]}$
 tells that $\delta(G)-1\leq \chi^{'}_{c}(G)\leq \delta(G)$.

  Let $f$ be a positive integer-valued function defined on $V$ such that
  $f(v)\leq d(v)$ for every vertex $v \in V$.  An $f$-edge
cover-colouring of $G$ is a colouring of edges of $G$  such
that every colour appears at each vertex $v \in V$ at least $f(v)$
times. The maximum number of colours needed for $f$-edge cover
colour $G$ is called the $f$-edge cover chromatic index
of $G$, denoted by $\chi^{'}_{fc}(G)$.

If there is a vertex $x\in V(G)$ with $\frac{d(x)}{2} d(x)$, then$\chi^{'}_{fc}(G)\leq 1$. In general, we suppose$f(v)\leq \frac{d(v)}{2}$for every vertex$v\in V$.

Key words: Edge colouring;edge cover-colouring, f-edge cover-colouring