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On edge colorings of 1-planar graphs without 5-cycles with two chords. (English) Zbl 1409.05091
A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. Let $$G$$ be a 1-planar graph of maximum degree $$\Delta$$. In this paper the following theorem is proved: If $$\Delta \geqslant 8$$ and any 5-cycle of $$G$$ has at most one chord, then $$G$$ is edge-colorable with $$\Delta$$ colors. This theorem weakens the hypotheses of a previous theorem of this type by X. Zhang and G. Z. Liu [Ars Comb. 104, 431–436 (2012; Zbl 1274.05186)]: If $$\Delta \geqslant 9$$ and any 5-cycle of $$G$$ has no chord at all, then $$G$$ is edge-colorable with $$\Delta$$ colors.
The authors use their “discharging method” which requires a case-by-case analysis based on V. G. Vizing’s adjacency lemma [Russ. Math. Surv. 23, No. 6, 125–141 (1968; Zbl 0192.60502); translation from Usp. Mat. Nauk 23, No. 6(144), 117–134 (1968)].
Notice that in the 1-planar case, the condition $$\Delta \geqslant 8$$ is not a necessary one; for example, the complete four-partite graph $$K_{2,2,2,2}$$ is 1-planar and is edge-colorable with $$\Delta = 6$$ colors.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C35 Extremal problems in graph theory 05C76 Graph operations (line graphs, products, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory
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