Abstract
Let G be a graph and e=uv an edge in G (also a vertex in the line graph L(G) of G). Then e is in two cliques EG(u) and EG(v) with EG(u)∩EG(v)={e} of L(G), that correspond to all edges incident with u and v in G respectively. Let SL(G) be any spanning subgraph of L(G) such that every vertex e=uv is adjacent to at least min{dG(u)−1,⌈34dG(u)+12⌉} vertices of EG(u) and to at least min{dG(v)−1,⌈34dG(v)+12⌉} vertices of EG(v). Then if L(G) is Hamiltonian, we show that SL(G) is Hamiltonian. As a corollary we obtain a lower bound on the number of edge-disjoint Hamiltonian cycles in L(G).
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