Journal of Combinatorial Optimization

, Volume 28, Issue 1, pp 274–287 | Cite as

Minimum number of disjoint linear forests covering a planar graph

Article

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 

References

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

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Huijuan Wang
    • 1
  • Lidong Wu
    • 2
  • Weili Wu
    • 2
    • 3
  • Jianliang Wu
    • 1
  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  3. 3.College of Computer Science and TechnologyTaiYuan University of TechnologyTaiyuanChina

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