Total coloring of outer-1-planar graphs with near-independent crossings

Xin Zhang ()
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Xin Zhang: Xidian University

Journal of Combinatorial Optimization, 2017, vol. 34, issue 3, No 1, 675 pages

Abstract: Abstract 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 $$|M_G(c_1)\cap M_G(c_2)|\le 1$$ | M G ( c 1 ) ∩ M G ( c 2 ) | ≤ 1 for any two distinct crossings $$c_1$$ c 1 and $$c_2$$ c 2 in G, where $$M_G(c)$$ 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 $$\Delta +1$$ Δ + 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.

Keywords: Outerplanar graph; Outer-1-planar graph; Local structure; Total coloring (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-016-0093-x

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