On the acyclic choosability of graphs

Abstract

A proper vertex coloring of a graphA graph is said to be acyclically k-choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k-choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. ß 2005 Wiley Periodicals, Inc. J Graph Theory 51: [281][282][283][284][285][286][287][288][289][290][291][292][293][294][295][296][297][298][299][300] 2006

A conjecture of Borodin and a coloring of Grünbaum

“…Whereas our coloring is quite similar to his, the analysis of its properties is far more involved. Before we proceed to the proof of Theorem 1, we mention that there has been a lot of recent research related to acyclic colorings and refer the reader, for example, to [2,3,[6][7][8]13,14]. The main questions considered in these articles concern acyclic colorings of planar graphs or graphs of given maximum degree, generalizations of acyclic colorings such as star colorings and list versions of the coloring results.…”

Acyclic choosability of planar graphs: a Steinberg like approach

“…Also, the relationship between the maximum average degree of G (or the girth of G) and its acyclic choice number was studied (see for example [9,8,5]). …”

The method of coloring in graphs and its application

“…More interesting results on colorings in graphs can be found in [41][42][43][44][45][46][47][48][49][50]. Finally, we present some problems for further research as follows.…”