数学物理学报(英文版) ›› 2005, Vol. 25 ›› Issue (1): 145-151.
宋慧敏,刘桂真
SONG Hui-Min, LIU Gui-Zhen
摘要:
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}
$f(v)\leq \frac{d(v)}{2}$ for every vertex $v\in V$.