# Minimum number of disjoint linear forests covering a planar graph

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## Abstract

Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree \(\varDelta \ge 7\). We have proved that the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \), if for each vertex \(v\in V(G)\), there are two integers \(i_v,j_v\in \{3,4,5,6,7,8\}\) such that any two cycles of length \(i_v\) and \(j_v\), which contain \(v\), are not adjacent. Clearly, if \(i_v=i, j_v=j\) for each vertex \(v\in V(G)\), then we can easily get one corollary: for two fixed integers \(i,j\in \{3,4,5,6,7,8\}\), if there is no adjacent cycles with length \(i\) and \(j\) in \(G\), then the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \).

## Keywords

Planar graph Linear forest Cycle Covering## Notes

### Acknowledgments

This work was supported in part by National Natural Science Foundation of China under grants 11201440, 11271006, 11301410 and Natural Science Basic Research Plan in Shaanxi Province of China under grant 2013JQ1002. This work was supported in part by Scientific Research Foundation for the Excellent Young and Middle-Aged Scientists of Shandong Province of China under grant BS2013DX002. This work was also supported in part by National Science Foundation of USA under grants CNS-0831579 and CCF-0728851.

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