Note to the paper of Grünbaum on acyclic colorings

On the acyclic choosability of graphs

“…To see that G is not acyclically 4-colorable, consider its four 4-vertices: Any two of them are either adjacent or have four common neighbors. Thus, different colors are assigned to We now use the construction proposed in [8] to obtain an asymptotic upper bound on f ðnÞ. Let G n be the graph defined as follows: G n is a ðn þ 1Þ-clique in which each edge is replaced by n paths with length 2 (see the graph G 3 depicted in Fig.…”

“…Moreover, there exist bipartite 2-degenerate planar graphs which are not acyclically 4-colorable [8] (see Fig. 1).…”

Star coloring bipartite planar graphs

“…This was confirmed by Borodin [2] in 1979. Kostochka and Melnikov [5] gave examples of a bipartite planar graph that requires five colors to acyclically color so that Borodin's result is optimal, even when restricted to bipartite planar graphs.…”

“…A corollary to a result of Nesetȓil and Ossona de Mendez [6] shows that any bipartite planar graph can be star colored with 18 colors, but the best published constructions of bipartite planar graphs require only 5 colors to star color: Fertin et al [3] establish this for grid graphs of size at least 4 × 4, whereas Ramamurthi and Sanders [7] show that bipartite outerplanar graphs can be 5-star colored and that this is best possible by exhibiting a small outerplanar gridlike graph requiring 5 colors. Of course the examples of Kostochka and Melnikov [5] also require at least 5 colors to star color: for example, the graph obtained from the double-wheel C 5 ∨K 2 by replacing each edge uv of the 5-cycle with a copy of K 2,4 in which u, v are the degree 4 vertices.…”

Star coloring planar graphs from small lists

“…In 1979, Borodin [2] confirmed Grünbaum's conjecture. This upper bound is best possible since there are planar graphs that require 5 colors to acyclically color (see [6] for example). Grünbaum noted that the star chromatic number of a graph is bounded by a function of its acyclic chromatic number.…”