Discrete Mathematics 308 (2008) 5856–5859
www.elsevier.com/locate/disc
On the size of edge-coloring critical graphs with maximum
degree 4
Lianying Miao, Shiyou Pang
School of Science, China University of Mining and Technology, Xuzhou, 280001, PR China
Received 29 May 2006; received in revised form 10 September 2007; accepted 4 October 2007
Available online 4 March 2008
Abstract
In 1968, Vizing proposed the following conjecture: If G = (V, E) is a ∆-critical graph of order n and size m, then
m ≥
1
2
[(∆ − 1)n + 3]. This conjecture has been verified for the cases of ∆ ≤ 5. In this paper, we prove that m ≥
7
4
n when
∆ = 4. It improves the known bound for ∆ = 4 when n > 6.
c
2008 Published by Elsevier B.V.
Keywords: Conjecture; Edge-coloring; Edge-coloring critical graphs
1. Introduction
In this paper, all graphs G = (V , E) are finite, simple and undirected. Throughout, G is assumed to have n vertices
and m edges. The chromatic index χ
(G) of a graph G is the minimum number of colors required to color the edges
of G so that two adjacent edges receive different colors. In 1965, Vizing [5] proved that if G is a graph of maximum
degree ∆, then the chromatic χ
(G) is either ∆ or ∆ + 1. A graph G is said to be of Class one if χ
(G) = ∆, and it
is said to be of Class two if χ
(G) = ∆ + 1. A ∆-critical graph G is a connected graph of maximum degree ∆ such
that G is of Class two and G − e is of Class one for each edge e of G. The following is a well known conjecture of
Vizing proposed in 1968.
Conjecture (Vizing [6]). If G = (V , E) is a ∆-critical graph, then m ≥
1
2
[(∆ − 1)n + 3].
The conjecture has been proved for the case ∆ ≤ 5 [1,6].
In [4], Sanders and Zhao proved that if G = (V, E ) is a ∆-critical graph, then
m ≤
1
4
n(∆ +
√
2∆ − 1).
For ∆ ∈ {6, 7, 8, 9, 10, 11}, Yue Zhao [7] also proved that if G = (V, E) is a ∆-critical graph, then m ≥
nd
∆
2
, where
E-mail address: miaolianying@cumt.edu.cn (L. Miao).
0012-365X/$ - see front matter
c
2008 Published by Elsevier B.V.
doi:10.1016/j.disc.2007.10.013