Star coloring bipartite planar graphs

Abstract

Abstract:A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color.

Star coloring planar graphs from small lists

“…One of y 2 or y 3 must be assigned color 2, and one of y 7 or y 8 must be assigned color 2. Regardless of where color 2 is assigned, y 5 is the center of a 2-cluster since each of y 2 , y 3 , y 7 and y 8 are second neighbors of y 5 . This shows that color 2 may not be assigned to y 5 .…”

“…Regardless of where color 2 is assigned, y 5 is the center of a 2-cluster since each of y 2 , y 3 , y 7 and y 8 are second neighbors of y 5 . This shows that color 2 may not be assigned to y 5 . Similarly c(y 9 ) = 2, so we must have c(y 6 ) = c(y 8 ) = 2.…”

“…In order to control the number of colors used in an in-coloring, it is useful to bound the maximum outdegree of the orientation G. For instance in [5], it is shown that any bipartite planar graph G has a star-coloring with 14 colors by using a specific orientation with maximum outdegree 2. Tarsi [8] showed that a graph has an orientation with maximum outdegree at most d if and only if mad(G) ≤ 2d, where mad(G) denotes the maximum possible average degree over all subgraphs of G. A planar graph G of girth g has mad(G) < 2g /(g−2), so it has an orientation of outdegree at most 2 as long as g ≥ 4.…”

“…The next section suggests a way of constructing such graphs. This "cluster method" was introduced in [9], and was used in [5] and [10].…”

“…Thus there are planar graphs of arbitrarily high girth that require 4 colors to star color. Recently Kierstead et al [5] showed that bipartite planar graphs can be (list) star colored with 14 colors, and gave an example of a bipartite planar graph that requires 8 colors to star color. For low girth planar graphs, the bounds proved in [1] and [5] are the best known.…”