Forbidden Subgraphs and the Hamiltonian Index of a $2$-Connected Graph

Abstract

Hamiltonian index of a graph $G$ is the smallest positive integer $k$, for which the $k$-th iterated line graph $L^k(G)$ is hamiltonian. Bedrossian characterized all pairs of forbidden induced subgraphs that imply hamiltonicity in $2$-connected graphs. In this paper, some upper bounds on the hamiltonian index of a $2$-connected graph in terms of forbidden not necessarily induced subgraphs are presented.