Some properties of kk-trees

Let k≥2k≥2 be an integer. We investigate Hamiltonian properties of kk-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 GG. The branch number is denoted by β(G)β(G), which is a measure of how complex the kk-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 kk-trees and show that the branch number increases for more complex kk-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)β(G) and other graph parameters. We then use our structural results to show that if β(G)