On forbidden subdivision characterizations of graph classes

We provide a characterization of several graph parameters (the acyclic chromatic number, the arrangeability, and a sequence of parameters related to the expansion of a graph) in terms of forbidden subdivisions. Let us start with several definitions. Throughout the paper, we consider only simple undirected graphs. A graph G = sdt(G) is the t-subdivision of a graph G, if G is obtained from G by replacing each edge by a path with exactly t inner vertices. Similarly, G is a ≤ t-subdivision of G if the graph G can be obtained from G by subdividing each edge by at most t vertices (the number of vertices may be different for each edge). A coloring of vertices of a graph G is proper if no two adjacent vertices have the same color. The minimum k such that the graph G has a proper coloring by k colors is called the chromatic number of G and denoted by χ(G). A proper coloring of a graph G is acyclic if the union of each two color classes induces a forest, i.e., there is no cycle colored by two colors. The minimum k such that the graph G has an acyclic coloring by k colors is called the acyclic chromatic number of G and denoted by χa(G). In this paper, we present an exact characterization of graph classes whose acyclic chromatic number is bounded by a constant. As a motivation, let us consider several older results. Borodin [2] have proved that the acyclic chromatic number of every planar graph is at most 5. Nešetřil and Ossona de Mendez [5] have proved that every graph G has a minor H such that Supported as project 1M0545 by the Czech Ministry of Education