Linear Arboricity of Outer-1-Planar Graphs

  • Xin ZhangEmail author
  • Bi Li


A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. Zhang et al. (Edge covering pseudo-outerplanar graphs with forests, Discrete Math 312:2788–2799, 2012; MR2945171) proved that the linear arboricity of every outer-1-planar graph with maximum degree \(\Delta \) is exactly \(\lceil \Delta /2\rceil \) provided that \(\Delta =3\) or \(\Delta \geqslant 5\) and claimed that there are outer-1-planar graphs with maximum degree \(\Delta =4\) and linear arboricity \(\lceil (\Delta +1)/2\rceil =3\). It is shown in this paper that the linear arboricity of every outer-1-planar graph with maximum degree 4 is exactly 2 provided that it admits an outer-1-planar drawing with crossing distance at least 1 and crossing width at least 2, and moreover, none of the above constraints on the crossing distance and crossing width can be removed. Besides, a polynomial-time algorithm for constructing a path-2-coloring (i.e., an edge 2-coloring such that each color class induces a linear forest, a disjoint union of paths) of such an outer-1-planar drawing is given.


Outer-1-planar graph Crossing Linear arboricity Polynomial-time algorithm 

Mathematics Subject Classification

05C10 05C15 


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Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina

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