Some properties of k-trees

Let k >= 2 be an integer. We investigate Hamiltonian properties of k-trees, a special family of chordal graphs. Instead of studying the toughness condition motivated by a conjecture of Chvatal, we introduce a new parameter, the branch number of G. The branch number is denoted by beta(G), which is a measure of how complex the k-tree is. For example, a path has only two leaves and is said to be simple when compared to a tree with many leaves and long paths. We generalize this concept to k-trees and show that the branch number increases for more complex k-trees. We will see by the definition that the branch number is easier to calculate and to work with than the toughness of a graph. We give some results on the relationships between beta (G) and other graph parameters. We then use our structural results to show that if beta(G) < k, then there is a Hamilton path between any pair of vertices that passes through a given set of edges. Using this result, we show that if beta(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. (C) 2010 Elsevier B.V. All rights reserved.