Total coloring of outer-1-planar graphs with near-independent crossings
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A graph G is outer-1-planar with near-independent crossings if it can be drawn in the plane so that all vertices are on the outer face and \(|M_G(c_1)\cap M_G(c_2)|\le 1\) for any two distinct crossings \(c_1\) and \(c_2\) in G, where \(M_G(c)\) consists of the end-vertices of the two crossed edges that generate c. In Zhang and Liu (Total coloring of pseudo-outerplanar graphs, arXiv:1108.5009), it is showed that the total chromatic number of every outer-1-planar graph with near-independent crossings and with maximum degree at least 5 is \(\Delta +1\). In this paper we extend the result to maximum degree 4 by proving that the total chromatic number of every outer-1-planar graph with near-independent crossings and with maximum degree 4 is exactly 5.
KeywordsOuterplanar graph Outer-1-planar graph Local structure Total coloring
The author appreciates the anonymous referees sincerely for their helpful comments. This paper is supported by National Natural Science Foundation of China (No. 11301410), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130203120021), and the Fundamental Research Funds for the Central Universities (No. JB150714).