Clark proved that L(G) is hamiltonian if G is a connected graph of order n 2 6 such that deg u + deg v 2 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 dn) 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 c...

Thomassen (J. Combin. Theory Ser. B 28, 1980, 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to nd such a v...

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

Given positive integers k £ m £ n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m £ r £ n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best pos...

We prove the following approximate version of Pósa’s theorem for directed graphs: every directed graph on n vertices whose inand outdegree sequences satisfy di ≥ i+o(n) and d+i ≥ i+o(n) for all i ≤ n/2 has a Hamilton cycle. In fact, we prove that such digraphs are pancyclic (i.e. contain cycles of lengths 2, . . . , n). We also prove an approximate version of Chvátal’s theorem for digraphs. Thi...

In this paper, we consider the intersection graph IΓ(Zn) of gamma sets in the total graph on Zn. We characterize the values of n for which IΓ(Zn) is complete, bipartite, cycle, chordal and planar. Further, we prove that IΓ(Zn) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivit...

The well-known Chvátal-Erdős theorem states that if the stability number α of a graph G is not greater than its connectivity then G is hamiltonian. In 1974 Erdős showed that if, additionally, the order of the graph is sufficiently large with respect to α, then G is pancyclic. His proof is based on the properties of cycle-complete graph Ramsey numbers. In this paper we show that a similar result...