Affiliation:
1. School of Information Science and Technology, Yunnan Normal University , Kunming 650500 , China
Abstract
Abstract
The two-distance vertex-distinguishing index
χ
d
2
′
(
G
)
{\chi }_{d2}^{^{\prime} }\left(G)
of graph
G
G
is defined as the smallest integer
k
k
, for which the edges of
G
G
can be properly colored using
k
k
colors. In this way, any pair of vertices at distance of two have distinct sets of colors. The two-distance vertex-distinguishing edge coloring of graphs can be used to solve some network problems. In this article, we used the method of discharging to prove that if
G
G
is a graph with mad
(
G
)
<
8
3
\left(G)\lt \frac{8}{3}
, then
χ
d
2
′
(
G
)
≤
max
{
7
,
Δ
+
2
}
{\chi }_{d2}^{^{\prime} }\left(G)\le \max \left\{7,\Delta +2\right\}
, which improves the result that a graph
G
G
of
Δ
(
G
)
≥
4
\Delta \left(G)\ge 4
has
χ
d
2
′
(
G
)
≤
Δ
(
G
)
+
2
{\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+2
if mad
(
G
)
<
5
2
\left(G)\lt \frac{5}{2}
, and
χ
d
2
′
(
G
)
≤
Δ
(
G
)
+
3
{\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+3
if mad
(
G
)
<
8
3
\left(G)\lt \frac{8}{3}
.