Hamilton-connected indices of graphs

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131-148] defined hc(G) to be the least integer m such that the iterated line graph L^m(G) is Hamilton-connected. Let diam(G) be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G. In this paper, we show that [email protected]?hc(G)@?max{diam(G),k-1}. We also show that @k^3(G)@?hc(G)@[email protected]^3(G)+2 where @k^3(G) is the least integer m such that L^m(G) is 3-connected. Finally we prove that hc(G)@?|V(G)|[email protected](G)+1. These bounds are all sharp.

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