We present (9 4 −)-tough graphs without a Hamilton path for arbitrary ¿ 0, thereby refuting a well-known conjecture due to Chvà atal. We also present (7 4 −)-tough chordal graphs without a Hamilton path for any ¿ 0.

For a fixed graph H, let H‐CON denote the problem of determining whether a given graph is contractible to H. The complexity of H‐CON is studied for H belonging to certain classes of graphs, together covering all connected graphs of order at most 4. In particular, H‐CON is NP‐complete if H is a connected triangle‐free graph other than a star. For each connected graph H of order at most 4 other than P4 and C4, H‐CON is solvable in polynomial time.

A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d(x) + d(y) + d(z) 3s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and (Y the cardinality of a maximum independent set of vertices. If G is l-tough and s an, then every longest cycle in G is a dominating cycle and c z min(n, n + fs-cu) >, min(n , $n + 4s) 3 &a. If G is 2-connected and s 2 n + 2, then also c 3 min(n, n + 4s-(u), generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s 2 n, then G is hamiltonian.

No abstract

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 reline 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(1;)>2(jn-1) for every edge uu of G, then, for n sufficiently large, L(G) is hamiltonian.0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDI 0012-365X(92)00063-A

A cycle C of a graph G is called dominating cycle (D-cycle) if every edge of G is incident with at least one vertex of C. A D-path is defined analogously. If a graph G contains a D-cycle (D-path), the n its edge graph t(G) has a hamiltonian cycle (hamiltonian path). Necessary conditions and sufficient conditions are obtained for graphs to have a D-cycle or D-path. They are analogous to known conditions for the existence of hamiltonian cycles or paths. The notions edge degree and remote edges arise as analogues of uettex degree and nccnadjacent uetices, respectively. A result of Nash-Williams is improved.

A well-known conjecture in hamiltonian graph theory states that every 2-tough graph is hamiltonian. We give some equivalent conjectures, e.g., the conjecture that every 2-tough graph is hamiltonianconnected.

Using a variation of the Bondy-Chvbtal closure theorem the following result is proved: If G is a 2-connected graph with n vertices and connectivity K such that d(x) + d(y) + d(z) 2 n + K for any triple of independent vertices x, y, z, then G is hamiltonian.