Contents
Research Article
A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture
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- Published online by Cambridge University Press:
- 20 November 2018, pp. 440-448
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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.
