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

Abstract

Abstract For a given list assignment L = { L ( v ) : v ∈ V ( G )}, a graph G = ( V , E ) is L -colorable if there exists a proper coloring c of G such that c ( v ) ∈ L ( v ) for all v ∈ V . If G is L -colorable for every list assignment L having | L ( v )| ≥ k for all v ∈ 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.