Journal of East China Normal University(Natural Sc ›› 2015, Vol. 2015 ›› Issue (1): 131-135.doi: 10.3969/j.issn.1000-5641.2015.01.016

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The linear arboricity of upper-embedded graph and secondary upper-embedded graph

LYU Chang-Qing   

  1. School of Mathematics and Statistics, Zaozhuang University, Zaozhuang Shandong, 277160, China
  • Received:2014-04-01 Online:2015-01-25 Published:2015-03-29

Abstract: The linear arboricity of a graph $G$ is the minimum number of linear forests which partition the edges of $G$. In the present, it is proved that if a upper-embedded graph $G$ has $\Delta\geqslant 3\sqrt{4-3\varepsilon}$  then its linear arboricity is $\lceil\frac{\Delta}{2}\rceil$\,and if a secondary upper-embedded graph $G$ has $\Delta\geqslant 6\sqrt{1-\varepsilon}$ then its linear arboricity is $\lceil \frac{\Delta}{2}\rceil$, where $\varepsilon\leqslant0$. It improves the bound of the conclusion in [1]. As its application, the linear arboricity of a  triangulation graph on double torus is concluded

Key words: linear arboricity, surface, (secondary) upper-embedded graph, Euler , characteristic

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