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 \(\Delta \ge 17\).
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Communicated by Xueliang Li.
This work is partially supported by SRFDP (No. 20130203120021), NSFC (No. 11301410), and the Fundamental Research Funds for the Central Universities (No. JB150714).
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Zhang, X. On Equitable Colorings of Sparse Graphs. Bull. Malays. Math. Sci. Soc. 39 (Suppl 1), 257–268 (2016). https://doi.org/10.1007/s40840-015-0291-1
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DOI: https://doi.org/10.1007/s40840-015-0291-1