Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a（G）, is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g（G）, and let 1≤r≤2Δ be an intege...

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Bibliographic Details
Published inScience China. Mathematics Vol. 55; no. 12; pp. 2593 - 2600
Main Authors Lin, QiZhong, Hou, JianFeng, Liu, Yue
Format Journal Article
LanguageEnglish
Published Heidelberg SP Science China Press 01.12.2012
Subjects
Alon
Applications of Mathematics
cgt
China
Coloring
Graphs
Integers
Mathematical analysis
Mathematics
Mathematics and Statistics
无环
着色图
系列平行图
色数
边染色
非循环
probability method
girth
acyclic edge coloring
series-parallel graphs
05C15
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Summary:A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a（G）, is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g（G）, and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c 〉 0 such that if g（G）≥cΔ r log（Δ2/r） then χa（G）≤Δ ＋ r ＋ 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a（G） = Δ if Δ≥4 and g（G）≥4; or Δ≥3 and g（G）≥5.
Bibliography:A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a（G）, is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g（G）, and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c 〉 0 such that if g（G）≥cΔ r log（Δ2/r） then χa（G）≤Δ ＋ r ＋ 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a（G） = Δ if Δ≥4 and g（G）≥4; or Δ≥3 and g（G）≥5.
LIN QiZhong, HOU JianFeng, LIU Yue（1College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China; 2Center for Discrete Mathematics, Fuzhou University, Fuzhou 350003, China）
11-1787/N
acyclic edge coloring, girth, probability method, series-parallel graphs
ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-012-4442-7