Abstract
The vertex arboricity \(\rho (G)\) of a graph \(G\) is the minimum number of colors to color \(G\) such that each color class induces a forest. The list vertex arboricity \(\rho _l(G)\) is the list-coloring version of this concept. Zhen and Wu conjectured that \(\rho _l(G)=\rho (G)\) whenever \(|V(G)|\le 3\rho (G)\). In this paper, we prove the weaker version of the conjecture obtained by replacing \(3\rho (G)\) with \(\frac{5}{2}\rho (G)+\frac{1}{2}\).
Similar content being viewed by others
References
Borodin, O.V., Ivanova, A.O.: Planar graphs without 4-cycles adjacent to 3-cycles are list vertex 2-arborable. J. Graph Theory 62, 234–240 (2009)
Borodin, O.V., Kostochka, A.V., Toft, B.: Variable degeneracy: extensions of Brooks and Gallais theorems. Discrete Math. 214, 101–112 (2000)
Chartrand, G., Kronk, H.V.: The point-arboricity of planar graphs. J. Lond. Math. Soc. 44, 612–616 (1969)
Chartrand, G., Kronk, H.V., Wall, C.E.: The point-arboricity of a graph. Isr. J. Math. 6, 169–175 (1968)
Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. Cong. Numer. 26, 125–157 (1979)
Noel, J.A.; Reed, B.A.; Wu, H.: A proof of a conjecture of Ohba (preprint). arXiv: 1211.1999 (2012)
Ohba, K.: On chromatic-choosable graphs. J. Graph Theory 40, 130–135 (2002)
Raspaud, A., Wang, W.: On the vertex-arboricity of planar graphs. Eur. J. Comb. 29, 1064–1075 (2008)
Reed, B., Sudakov, B.: List colouring when the chromatic number is close to the order of the graph. Combinatorica 25, 117–123 (2004)
Vizing, V.G.: Coloring the vertices of a graph in prescribed colors. Diskret Analiz 29, 3–10 (1976)
Xue, N., Wang, W.: The list point arboricity of some complete multi-partite graphs. Ars Comb. 105, 457–462 (2012)
Xue, N., Wu, B.: List point arboricity of graphs. Discrete Math. Algorithms Appl. 4, 1250027 (2012)
Zhen, L., Wu, B.: List point arboricity of dense graphs. Graphs Comb. 25, 123–128 (2009)
Acknowledgments
The authors are grateful to the referees for their careful reading and valuable comments. This work is supported by NSFC (11161046), Xinjiang Young Talent Project (2013721012).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, W., Wu, B., Yan, Z. et al. A Weaker Version of a Conjecture on List Vertex Arboricity of Graphs. Graphs and Combinatorics 31, 1779–1787 (2015). https://doi.org/10.1007/s00373-014-1466-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1466-5