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∈V, let Wcol2[G,L,v] be the set of vertices w∈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 wcol2(G) of G is the least k such that there is a total ordering L on V with |Wcol2[G,L,v]|≤k for all vertices v∈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.