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

摘要

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