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On Equitable Colorings of Sparse Graphs

  • Xin ZhangEmail author
Article

Abstract

A graph is equitably k-colorable if G has a proper vertex k-coloring such that the sizes of any two color classes differ by at most one. Chen, Lih and Wu conjectured that any connected graph G with maximum degree \(\Delta \) distinct from the odd cycle, the complete graph \(K_{\Delta +1}\) and the complete bipartite graph \(K_{\Delta ,\Delta }\) are equitably m-colorable for every \(m\ge \Delta \). Let \({\mathcal {G}}_k\) be the class of graphs G such that \(e(G')\le k (v(G')-2)\) for every subgraph \(G'\) of G with order at least 3. In this paper, it is proved that any graph in \({\mathcal {G}}_4\) with maximum degree \(\Delta \ge 17\) is equitably m-colorable for every \(m\ge \Delta \). As corollaries, we confirm Chen–Lih–Wu Conjecture for 1-planar graphs, 3-degenerate graphs and graphs with maximum average degree less than 6, provided that \(\Delta \ge 17\).

Keywords

Equitable coloring 3-Degenerate graph Average degree 1-Planar graph 

Mathematics Subject Classification

05C15 

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Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina

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