# Minimum number of disjoint linear forests covering a planar graph

## 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.

## References

- A\(\ddot{\rm I}\)t-djafer H (1987) Linear arboricity for graphs with multiple edges. J Graph Theory 11:135–140Google Scholar
- Akiyama J, Exoo G, Harary F (1980) Covering and packing in graphs III: cyclic and acyclic invariants. Math Slovaca 30:405–417zbMATHMathSciNetGoogle Scholar
- Akiyama J, Exoo G, Harary F (1981) Covering and packing in graphs IV: Linear arboricity. Networks 11:69–72CrossRefzbMATHMathSciNetGoogle Scholar
- Alon N (1988) The linear arboricity of graphs. Israel J Math 62:311–325CrossRefzbMATHMathSciNetGoogle Scholar
- Alon N, Teague VJ, wormald NC (2001) Linear arboricity and linear \(k\)-arboricity of regular graphs. Graphs Combin 17:11–16CrossRefzbMATHMathSciNetGoogle Scholar
- Angelini P, Frati F (2012) Acyclically 3-colorable planar graphs. J Comb Optim 24:116–130CrossRefzbMATHMathSciNetGoogle Scholar
- Bessy S, Havet F (2013) Enumerating the edge-colourings and total colourings of a regular graph. J Comb Optim 25:523–535CrossRefzbMATHMathSciNetGoogle Scholar
- Bondy JA, Murty USR (1976) Graph theory with applications (M). North-Holland, New YorkGoogle Scholar
- Chen HY, Tan X, Wu JL. The linear arboricity of planar graphs with maximum degree at least 7, Utilitas Math, to appearGoogle Scholar
- Cygan M, Hou JF, Kowalik L, Luzar B, Wu JL (2012) A planar linear arboricity conjecture. J Graph Theory 69:403–425CrossRefzbMATHMathSciNetGoogle Scholar
- Du HW, Jia XH, Li DY, Wu WL (2004) Coloring of double disk graphs. J Global Optim 28:115–119CrossRefzbMATHMathSciNetGoogle Scholar
- Enomoto H, Péroche B (1984) The linear arboricity of some regular graphs. J Graph Theory 8:309–324CrossRefzbMATHMathSciNetGoogle Scholar
- Guldan F (1986) The linear arboricity of 10 regular graphs. Math Slovaca 36:225–228zbMATHMathSciNetGoogle Scholar
- Harary F (1970) Covering and packing in graphs I. Ann NY Acad Sci 175:198–205CrossRefzbMATHGoogle Scholar
- Li XW, Mak-Hau V, Zhou SM (2013) The \(L(2,1)\)-labelling problem for cubic cayley graphs on dihedral groups. J Comb Optim 25:716–736CrossRefzbMATHMathSciNetGoogle Scholar
- McDiarmid CJH, Reed BA (1990) The linear arboricity of random regular graphs. Random Struct Algorithms 1:443–445CrossRefzbMATHMathSciNetGoogle Scholar
- Péoche B (1982) Complexity of the linear arboricity of a graph. RAIRO Rech Oper 16:125–129MathSciNetGoogle Scholar
- Wang WF, Wang YQ (2010) Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree. J Comb Optim 19:471–485CrossRefzbMATHMathSciNetGoogle Scholar
- Wu JL (1999) On the linear arboricity of planar graphs. J Graph Theory 31:129–134CrossRefzbMATHMathSciNetGoogle Scholar
- Wu JL, Wu YW (2008) The linear arboricity of planar graphs of maximum degree seven are four. J Graph Theory 58:210–220CrossRefzbMATHMathSciNetGoogle Scholar
- Wu JL (1996) Some path decompositions of Halin graphs. J Shandong Mining Inst 15:219–222Google Scholar
- Wu JL (2000) The linear arboricity of series-parallel graphs. Graphs Combin 16:367–372CrossRefzbMATHMathSciNetGoogle Scholar
- Wu JL, Hou JF, Liu GZ (2007) The linear arboricity of planar graphs with no short cycles. Theor Comp Sci 381:230–233CrossRefzbMATHMathSciNetGoogle Scholar
- Wu JL, Hou JF, Sun XY (2009) A note on the linear arboricity of planar graphs without 4-cycles. ISORA’09, 174–178Google Scholar