Minimum choosability of planar graphs

Abstract

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring $\phi$ such that $\phi (x) \in L(x)$. Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring $\phi$ such that $\phi (x) \in L(x)$. We proved $\chi '_{l}(G)=\Delta$ and $\chi ''_{l}(G)=\Delta +1$ for a planar graph G with maximum degree $\Delta \ge 8$ and without chordal 6-cycles, where the list edge chromatic number $\chi '_{l}(G)$ of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number $\chi ''_{l}(G)$ of G is the smallest integer k such that G is total-k-choosable.