On the s-hamiltonianicity of an hourglass-free line graph

Abstract

Fault-tolerant networks are often modeled as s-hamiltonian graphs. Thus it is of interests to find graph families in which whether a graph is s-hamiltonian can be determined in polynomial time. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryjáček and Vrána in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1-Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line graph L(G), each of the following holds.(i) If s≥2, then L(G) is s-hamiltonian if and only if κ(L(G))≥s+2;(ii) If s≥1, then L(G) is s-Hamilton-connected if and only if κ(L(G))≥s+3.