Hamiltonian Spectra of Graphs

Abstract

A hamiltonian walk in a digraph D is a closed spanning directed walk of D with minimum length. The length of a hamiltonian walk in D is called the hamiltonian number of D, and is denoted by h(D). The hamiltonian spectrum $$S_h(G)$$ of a graph G is the set $$\{h(D): D$$ is a strongly connected orientation of $$G\}$$ . In this paper, we present necessary and sufficient conditions for a graph G of order n to have $$S_h(G)=\{n\}$$ , $$\{n+1\}$$ , or $$\{n+2\}$$ . Then we construct some 2-connected graphs of order n with hamiltonian spectrum being a singleton $$n+k$$ for some $$k\ge 3$$ , and graphs with their hamiltonian spectra being sets of consecutive integers.