Hamiltonian cycles in spanning subgraphs of line graphs
Let G G be a graph and e=uv e = u v an edge in G G (also a vertex in the line graph L(G) L ( G ) of G G ). Then e e is in two cliques EG(u) E G ( u ) and EG(v) E G ( v ) with EG(u)?EG(v)={e} E G ( u ) ? E G ( v ) = { e } of L(G) L ( G ) , that correspond to all edges incident with u u and v v in G G respectively. Let SL(G) S L ( G ) be any spanning subgraph of L(G) L ( G ) such that every vertex e=uv e = u v is adjacent to at least m i n { d G ( u ) ? 1 , ? 3 4 d G ( u ) + 1 2 ? } vertices of EG(u) E G ( u ) and to at least m i n { d G ( v ) ? 1 , ? 3 4 d G ( v ) + 1 2 ? } vertices of EG(v) E G ( v ) . Then if L(G) L ( G ) is Hamiltonian, we show that SL(G) S L ( G ) is Hamiltonian. As a corollary we obtain a lower bound on the number of edge-disjoint Hamiltonian cycles in L(G) L ( G ) .
hamiltonian cycle spanning subgraph line graph