On the Hajo's number of graphs

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of Kdm=2e+1;bm=2c+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¿cn for some positive constant c. In fact, we prove that under the condition m¿cn, 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 Haj os number h(H)¿n, that is, with a subdivision of Kn+1. In addition, we prove bounds of Nordhaus–Gaddum type for the Haj os number. c © 2000 Elsevier Science B.V. All rights reserved.