Some properties of k-trees

Abstract. Let k ≥ 2 be an integer. We investigate hamiltonian properties for a k-tree G, a special chordal graph. Instead of studying the toughness condition motivated by a conjecture of Chvátal, we introduce a new parameter, the branch number of G, denoted by β(G). Some results on the relationships between β(G) and other graph parameters will be presented. A path system of G is a subgraph whose components are paths with length at least 1. One of our main results shows that if β(G) < k and T is any path system with at most k − (β(G) + 1) edges, then for any x and x not in the same path in T with dT (x) ≤ 1 and dT (x ) ≤ 1, we can find a hamiltonian (x, x)-path passing through T . Using this result, we show that if β(G) ≤ k, then G is hamiltonian. This generalizes a recent result of Broersma et al. which says that any k+1 3 -tough k-tree is hamiltonian.