On the Hajós number of graphs Original Research Article

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of K⌈m/2⌉+1,⌊m/2⌋+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that each of these subsets induces a graph with property Pn, provided m⩾n⩾1 (respectively, m⩾n⩾2). We prove that both conjectures fail when m>cn2 for some positive constant c. In fact, we prove that under the condition m>cn2, there exists a graph G with property Pm such that in every colouring of its vertices or edges with m colours there is a monochromatic subgraph H with Hajós number h(H)>n, that is, with a subdivision of Kn+1. In addition, we prove bounds of Nordhaus–Gaddum type for the Hajós number.