A Planar linear arboricity conjecture

作者：Cygan, Marek^*; Hou, Jian-Feng; Kowalik, Lukasz; Luzar, Borut; Wu, Jian-Liang

来源：Journal of Graph Theory, 2012, 69(4): 403-425.

DOI：10.1002/jgt.20592

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The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree ? is either ??/2? or ?(? + 1)/2?. In [J. L. Wu, J Graph Theory 31 (1999), 129134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210220], it was proven that LAC holds for all planar graphs. LAC implies that for ? odd, la(G) = ??/2?. We conjecture that for planar graphs, this equality is true also for any even ??6. In this article we show that it is true for any even ??10, leaving open only the cases ? = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when ??9.

• 出版日期2012-4
• 单位福州大学; 山东大学