An
f-coloring of a graph
G is an edge-coloring of
G such that each color appears at each vertex
v ∈
V(
G) at most
f(
v) times. The minimum number of colors needed to
f-color
G is called the
f-chromatic index of
G, and denoted by
χ′
f
(
G). Any simple graph
G has
f-chromatic index equal to
Δ
f
(
G) or
Δ
f
(
G) + 1, where
Df(G)=maxv Î V(G){é\fracd(v)f(v)ù}\Delta_{f}(G)=\max_{v\in V(G)}\{\lceil\frac{d(v)}{f(v)}\rceil\}
. If
χ′
f
(
G) =
Δ
f
(
G), then
G is of
f-class 1; otherwise
G is of
f-class 2. In this paper, we show that if
f(
v) is positive and even for all
v Î V0*(G)ÈNG(V0*(G))v\in V_0^*(G)\cup N_G(V_0^*(G))
, then
G is of
f-class 1, where
V*0(G)={v Î V(G):\fracd(v)f(v)=Df(G)}V^{*}_{0}(G)=\{v\in V(G):\frac{d(v)}{f(v)}=\Delta_{f}(G)\}
and
NG(V0*(G))={v Î V(G):uv Î E(G), u Î V0*(G)}N_G(V_0^*(G))=\{v\in V(G):uv\in E(G), u\in V_0^*(G)\}
. This result improves the simple graph version of a result of Hakimi and Kariv [4].
Keywords Edge-coloring - simple graph -
f-coloring -
f-chromatic index
This research is supported by NSFC(10471078, 60673047) and RSDP(20040422004) and NSF of Hebei(A2007000002) of China.