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A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture

Published online by Cambridge University Press:  20 November 2018

Haihui Zhang
Haihui Zhang*
Affiliation:
School of Mathematical Science, Huaiyin Normal University, 111 Changjiang West Road, Huaian, Jiangsu, 223300, P. R. China e-mail: hhzh@hytc.edu.cn
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Abstract

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For a given list assignment $L\,=\,\{L\left( v \right)\,:\,v\,\in \,V\left( G \right)\}$, a graph $G\,=\,\left( V,\,E \right)$ is $L$-colorable if there exists a proper coloring $c$ of $G$ such that $c\left( v \right)\,\in \,L\left( v \right)$ for all $v\,\in \,V$. If $G$ is $L$-colorable for every list assignment $L$ having $\left| L\left( v \right) \right|\,\ge \,k$ for all $v\,\in \,V$, then $G$ is said to be $k$-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.

Keywords

05C15choosabilityplanar graphcycle
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 440 - 448
Copyright
Copyright © Canadian Mathematical Society 2016

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