List Dynamic 4-Coloring of Planar Graphs

For a list assignment L of a graph G, a dynamicL-coloring of G is a proper vertex L-coloring of G such that for each vertex u of degree at least 2, the set N(u) of neighbors of u is not monochromatic. For positive integers k, s and t, a (k, s, t)-list assignment of a plane graph G is a list assignment L such that (i) |L(v)|>= k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|L(v)|\ge k$$\end{document} for each vertex v of G, (ii) |L(x)boolean AND L(y)|<= s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|L(x)\cap L(y)|\le s$$\end{document} for each edge xy of G, and (iii) |L(v)boolean AND L(w)|<= t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|L(v)\cap L(w)|\le t$$\end{document} if there is a vertex u such that v, u, w are three consecutive vertices on the boundary of a face. We say that a plane graph G is dynamically (k, s, t)-choosable if G is dynamically L-colorable for any (k, s, t)-list assignment L of G. We show that every plane graph is dynamically (4, 2, 2)-choosable. We also show that & Scaron;krekovski's conjecture about the (3,1)-choosability is equivalent to the statement that all plane graphs are dynamically (3,1,1)-choosable.