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.

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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