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A Structure of 1-Planar Graph and Its Applications to Coloring Problems

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Abstract

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the (p, 1)-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p, 1)-total labelling number of every 1-planar graph G is at most \(\Delta (G)+2p-2\) provided that \(\Delta (G)\ge 8p+2\) and \(p\ge 2\), and show that every 1-planar graph has an equitable edge coloring with k colors for any integer \(k\ge 18\). These three results respectively generalize the main theorems of three different previously published papers.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, China

    Xin Zhang & Bei Niu

  2. School of Information Science and Engineering, Qufu Normal University, Rizhao, 276826, Shandong, China

    Jiguo Yu

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  1. Xin Zhang
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  2. Bei Niu
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  3. Jiguo Yu
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Correspondence to Xin Zhang.

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This work was supported by the Fundamental Research Funds for the Central Universities (No. JB170706), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JM1010), and the National Natural Science Foundation of China (Nos. 11871055, 11301410).

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Zhang, X., Niu, B. & Yu, J. A Structure of 1-Planar Graph and Its Applications to Coloring Problems. Graphs and Combinatorics 35, 677–688 (2019). https://doi.org/10.1007/s00373-019-02027-0

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  • DOI: https://doi.org/10.1007/s00373-019-02027-0

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