A counterexample to a conjecture of Jackson and Wormald

A counterexample is presented to the following conjecture of Jackson and Wormald: If j ~ 1, k ~ 2 and a graph is connected, locally jconnected and K l , (j+l)(k-l)+2 -free then it has a k-tree. Preliminaries All graphs considered here are finite and without loops or multiple edges. As usual, we let V(G) and E(G) denote respectively the vertex set and the edge set of the graph G. The cardinality of the set 8 is denoted by 181. A K l ,k -free graph is a graph containing no copy of K l,k as an induced subgraph. Also, a graph is locally j-connected if every subgraph induced by the set of neighbours of a vertex v is j-connected. A k-tree of a graph is a spanning tree with maximum degree at most k. The join of two disjoint graphs G l and G2 , denoted by G l + G2 , is obtained by joining each vertex of G l to each vertex of G2 • The union of m disjoint copies of the same graph G is denoted by mG. In [1], Bill Jackson and Nicholas C. Wormald made the following conjecture: If j ~ 1, k ~ 2 and a graph is connected, locally j-connected and K l , (j+l)(k-l)+2 -free then it has a k-tree. A counterexample For any integers <5 ~ 2 and k ~ 2, first we construct the graph Gl + G21 where Gl = Ko and G2 = {<5(k 1) + I}Ko. Then join a K o(k-l) to each copy of Ko in G2 ; the graph is depicted in Figure 1. Australasian Journal of Combinatorics 19(1999), pp.259-260 ~8(k-lj II 6(k 1) + 1 copies