On the weak 2-coloring number of planar graphs

For a graph G = (V, E), a total ordering L on V, and a vertex v is an element of V, let Wcol(2)[G, L, v] be the set of vertices w is an element of V for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number wcol(2)(G) of G is the least k such that there is a total ordering L on V with | Wcol(2)[G, L, v]| <= k for all vertices v is an element of V. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved. (c) 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).