Discussiones Mathematicae Graph Theory 29(3) (2009) 499-510
doi: 10.7151/dmgt.1460

[BIBTex] [PDF] [PS]

THE LIST LINEAR ARBORICITY OF PLANAR GRAPHS

Xinhui An  and  Baoyindureng Wu

College of Mathematics and System Science
Xinjiang University
Urumqi 830046, P.R. China
e-mail: xjaxh@xju.edu.cn,  baoyin@xju.edu.cn

Abstract

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ≥ 13, or for any planar graph with Δ≥ 7 and without i-cycles for some i ∈ {3,4,5 }. We also prove that ⌈½Δ(G)⌉≤ lla(G)≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ≥ 9.

Keywords: list coloring, linear arboricity, list linear arboricity, planar graph.

2000 Mathematics Subject Classification: 05C10, 05C70.

References

[1] J. Akiyama, G. Exoo and F. Harary, Covering and packing in graphs III: Cyclic and acyclic invariants, Math. Slovaca 30 (1980) 405-417.
[2] J. Akiyama, G. Exoo and F. Harary, Covering and packing in graphs IV: Linear arboricity, Networks 11 (1981) 69-72, doi: 10.1002/net.3230110108.
[3] X. An and B. Wu, List linear arboricity of series-parallel graphs, submitted.
[4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, Macmillan, London, 1976).
[5] O.V. Borodin, On the total colouring of planar graphs, J. Reine Angew. Math. 394 (1989) 180-185, doi: 10.1515/crll.1989.394.180.
[6] H. Enomoto and B. Péroche, The linear arboricity of some regular graphs, J. Graph Theory 8 (1984) 309-324, doi: 10.1002/jgt.3190080211.
[7] F. Guldan, The linear arboricity of 10 regular graphs, Math. Slovaca 36 (1986) 225-228.
[8] F. Harary, Covering and packing in graphs I, Ann. N.Y. Acad. Sci. 175 (1970) 198-205, doi: 10.1111/j.1749-6632.1970.tb56470.x.
[9] J.L. Wu, On the linear arboricity of planar graphs, J. Graph Theory 31 (1999) 129-134, doi: 10.1002/(SICI)1097-0118(199906)31:2<129::AID-JGT5>3.0.CO;2-A.
[10] J.L. Wu, The linear arboricity of series-parallel graphs, Graphs Combin. 16 (2000) 367-372, doi: 10.1007/s373-000-8299-9.
[11] J.L. Wu, J.F. Hou and G.Z. Liu, The linear arboricity of planar graphs with no short cycles, Theoretical Computer Science 381 (2007) 230-233, doi: 10.1016/j.tcs.2007.05.003.
[12] J.L. Wu and Y.W. Wu, The linear arboricity of planar graphs of maximum degree seven is four, J. Graph Theory 58 (2008) 210-220, doi: 10.1002/jgt.20305.

Received 12 February 2008
Revised 12 January 2009
Accepted 28 April 2009