Every planar graph has an acyclic 7-coloring

A conjecture of Borodin and a coloring of Grünbaum

“…This result was originally conjectured by Grünbaum [9] in 1973 who proved the existence of acyclic 9-colorings for planar graphs. Borodin's proof-which was qualified as a major achievement in [10]-marked the end of a sequence of improvements of Grünbaum's original result by Mitchem [12], Albertson and Berman [1], and Kostochka [11]. Whereas Grünbaum's proof relied on a suitable decomposition of the planar graph into a sequence of outerplanar graphs and explicitely constructed the desired coloring using acyclic 3-colorings of these outerplanar graphs, the later improvements used the method of reducible configurations (up to 450) and discharging just like the proofs of the four color theorem.…”

Acyclic 3-coloring of generalized Petersen graphs

“…He proved that any planar graph is acyclically 9-colorable, and conjectured that five colors are sufficient. Since then there appeared many papers that investigated the upper bound of the acyclic chromatic number of planar graphs (Albertson and Berman 1977;Kostochka 1976;Mitchem 1974). In 1976, Borodin gave a proof of Grünbaum's conjecture by showing that every planar graph is acyclically 5-colorable.…”

On the acyclic choosability of graphs

“…The acyclic chromatic number of G, a ðGÞ, is the smallest integer k such that G is acyclically k-colorable. Acyclic colorings were introduced by Grünbaum in [17] and studied by Mitchem [11], Albertson, and Berman [1], and Kostochka [9]. In 1979, Borodin proved Grünbaum's conjecture:…”