Totally odd subdivisions and parity subdivisions: Structures and Coloring

A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H. In this paper, we give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is either 1. after deleting bounded number of vertices, an “almost” embedded graph into a bounded-genus surface, or 2. after deleting bounded number of vertices, a bipartite graph, or 3. after deleting bounded number of vertices, a graph with maximum degree at most f(|H|) for some function f of |H| (or a 6|H|-degenerate graph). Moreover, we can obtain either a totally odd Kksubdivision or such a tree-decomposition in polynomial time. We note that for minor-free graphs, we just need the first structure [37], while for odd-minor-free graphs, we need the first two structures [10]. For subdivisionfree graphs, we need the first and the third structures [17, 29]. So our result can be viewed as a combination of odd-minor-free graphs and subdivision-free graphs. The same conclusion of the structure theorem is true if we replace “totally odd” by “parity”. Hence this generalizes the structure theorem for subdivision-free graphs [17, 29]. We also consider coloring of graphs with no totally odd Kk-subdivision. We prove that any graph with no totally odd Kk-subdivision is 79k /4-colorable. The ∗National Institute of Informatics and JST ERATO Kawarabayashi Project, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo,