The Hamiltonian Number of Cubic Graphs

Abstract

A Hamiltonian walk in a connected graph G of order n is a closed spanning walk of minimum length in G. The Hamiltonian number h(G) of a connected graph G is the length of a Hamiltonian walk in G. Thus h may be considered as a measure of how far a given graph is from being Hamiltonian. We prove that if G runs over the set of connected cubic graphs of order n and $n\not= 14$ then the values h(G) completely cover a line segment [a,b] of positive integers. For an even integer n ≥ 4, let $\mathcal{C}(3^n)$ be the set of all connected cubic graphs of order n. We define $\min(h, 3^n)=\min\{h(G):G\in \mathcal{C}(3^n)\}$ and $\max(h, 3^n)=\max\{h(G):G\in \mathcal{C}(3^n)\}$. Thus for an even integer n ≥ 4, the two invariants min (h, 3ⁿ) and max (h, 3ⁿ) naturally arise. Evidently, min (h, 3ⁿ) = n. The exact values of max (h, 3ⁿ) are obtained in all situations.