Advertisement

Linear Arboricity of Outer-1-Planar Graphs

  • Xin ZhangEmail author
  • Bi Li
Article
  • 12 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

05C10 05C15 

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, New York (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Harary, F.: Covering and packing in graphs I. Ann. N. Y. Acad. Sci. 175, 198–205 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs III: cyclic and acyclic invariants. Math. Slovaca 30, 405–417 (1980)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Wu, J.L.: On the linear arboricity of planar graphs. J. Graph Theory 31, 129–134 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wu, J.L., Wu, Y.: The linear arboricity of planar graphs of maximum degree seven are four. J. Graph Theory 58, 210–220 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cygan, M., Hou, J., Kowalik, Ł., Lužar, B., Wu, J.L.: A planar linear arboricity. J. Graph Theory 69(4), 403–425 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wu, J.L.: The linear arboricity of series–parallel graphs. Graphs Comb. 16, 367–372 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Peroche, B.: Complexity of the linear arboricity of a graph. RAIRO Oper. Res. 16, 125–129 (1982). In FrenchMathSciNetCrossRefGoogle Scholar
  9. 9.
    Eggleton, R.B.: Rectilinear drawings of graphs. Util. Math. 29, 149–172 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Zhang, X., Liu, G., Wu, J.L.: Edge covering pseudo-outerplanar graphs with forests. Discrete Math. 312, 2788–2799 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhang, X.: List total coloring of pseudo-outerplanar graphs. Discrete Math. 313, 2297–2306 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Auer, C., Bachmaier, C., Brandenburg, F.J., et al.: Outer-1-planar graphs. Algorithmica 74, 1293–1320 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dehkordi, H.R., Eades, P.: Every outer-1-plane graph has a right angle crossing drawing. Int. J. Comput. Geom. Appl. 22, 543–557 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Auer, C., Bachmaier, C., Brandenburg, F.J., et al.: Recognizing outer 1-planar graphs in linear time. Lect. Notes Comput. Sci. 8242, 107–118 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hong, S.-H., Eades, P., Katoh, N., et al.: A linear-time algorithm for testing outer-1-planarity. Algorithmica 72, 1033–1054 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations