Some properties of -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 Chvátal, we introduce a new parameter, the branch number of G . The branch number is denoted by β ( 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 β ( G ) and other graph parameters. We then use our structural results to show that if β ( 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 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.