Abstract
Let \(G=(V(G),E(G))\) be a graph and s, t integers with \(s\le t\). If we can assign an s-subset \(\phi (v)\) of the set \(\{1, 2,\ldots ,t\}\) to each vertex v of V(G) such that \(\phi (u)\cap \phi (v)=\emptyset \) for every edge \(uv\in E(G)\), then G is called (t : s)-colorable, and such an assignment \(\phi \) is called a (t : s)-coloring of G. Let \(C_n\) denote a cycle of length n. In this paper, we show that every planar graph without \(C_4\) and \(C_6\) is (7 : 2)-colorable and thus has fractional chromatic number at most 7/2.
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions. CHEN is partially supported by NSFC under Grant Nos. 11671198, 11871270 and 11931006. HU is partially supported by NSFC under Grant Nos. 11601176 and 11971196.
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Wu, H., Chen, Y. & Hu, X. Planar graphs without 4- and 6-cycles are (7 : 2)-colorable. J Comb Optim 40, 45–58 (2020). https://doi.org/10.1007/s10878-020-00571-7
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DOI: https://doi.org/10.1007/s10878-020-00571-7