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

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.

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