Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n - 1 - p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n - 1 - p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G)  2 and deg u + deg v ≥ n - 1 for every pair of nonadjacent vertices u and v. The bound n - 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.
['Abdelhamid Benhocine', 'Louise N. Clark', 'Norbert Kohler', 'H. Veldman']
I am writing a research paper on a idea On circuits and pancyclic line graphs
[{"from": "human", "value": "I am writing a research paper on a idea On circuits and pancyclic line graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Clark proved that L(G) is hamiltonian if G is a connected graph of order n \u2265 6 such that deg u + deg v \u2265 n - 1 - p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n - 1 - p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n \u2265 6, \u03b4(G)  2 and deg u + deg v \u2265 n - 1 for every pair of nonadjacent vertices u and v. The bound n - 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail."}]