A result on Hamiltonian line graphs involving restrictions on induced subgraphs

It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph

On dominating and spanning circuits in graphs

An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was made by Catlin, who defined the reduction of a graph G and showed that G has a spanning circuit if and only if the reduction of G has a spanning circuit. We refine Catlin's reduction technique to obtain a result which contains several known and new sufficient conditions for a graph to have a spanning or dominating circuit in terms of degree-sums of adjacent vertices. In particular, the result implies the truth of the following conjecture of Benhocine et al.: If G is a connected simple graph of order n such that every cut edge of G is incident with a vertex of degree 1 and d(u)+d(v) > 2(1/5n-1) for every edge uv of G, then, for n sufficiently large, L(G) is hamiltonian