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This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs. Corollary 3 [4] Let G be a graph of order n with δ(G) ≥ (2n − 1)/3 and suppose n ≥ n 1 + n 2 +. .. + n k where n i ≥ 3 for all i. Then G contains the vertex disjoint union of the cycles C n 1 ∪ C n 2 ∪. .. ∪ C n k. Clearly then, any such graph contains any 2-factor we would want and hence provides a strong analogue to Dirac's theorem. I should mention that Alon and Fischer [6] also provided a solution to the Sauer-Spencer conjecture (δ = 2n/3). Their result used work dependent on the regularity lemma and thus holds only for large graphs. Related to the last result is another old conjecture due to El-Zahar [95]. Conjecture 2 Let G be a graph of order n = n 1 + n 2 +. .. + n k with δ(G) ≥ k i=1 ⌈n i /2⌉, then G contains the 2-factor C n 1 ∪. .. ∪ C n k. Note that the graph K s−1 + K ⌈ n−s+1 2 ⌉,⌈ n−s+1 2 ⌉ has minimum degree (n + s − 1)/2 but contains no s vertex disjoint odd length cycles. Thus, the conjecture is best possible. El-Zahar [95] provided an affirmative answer to the case k = 2, while Dirac's Theorem handles k = 1. Recently, Abbasi [1] announced a solution for large n using the regularity lemma. It would still be interesting to find a solution to this beautiful conjecture for all n. It should be noted that Corrádi and Hajnal [87] provided an affirmative answer to the El-Zahar conjecture for the case that each n i = 3 An old conjecture of Erdös and Faudree [102] generalizes the Corrádi-Hajnal theorem in another direction. Conjecture 3 Let G be a graph with order n = 4k and δ(G) ≥ 2k, then G contains k vertex disjoint 4-cycles. Alon and Yuster [8] proved that for any ǫ > 0, there exists k 0 such that if G has order 4k and δ(G) ≥ (2 + ǫ)k with k ≥ k 0 , then G contains k disjoint 4-cycles. In [207], a near solution was provided by Randerath, Schiermeyer and Wang.

An assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) c 1 n /, for graphs with maximum degree Á n 1=2 and minimum ------------------

It is known that for a graph on n vertices n 2 /4 + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex.

This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, harniltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area. INTRODUCTIONThe hamiltonian problem; determining when a graph contains a spanning cycle has long been fundamental in graph theory. Named for Sir William Rowan Hamilton, this problem traces its origins to the 1850s. Today, however, the flood of papers dealing with this subject and its many related problems is at its greatest; supplying us with new results as well as many new problems involving cycles and paths in graphs.To many, including myself, any path or cycle question is really a part of this general area. Although it is difficult to separate many of these ideas, for the purpose of this article, I will concentrate my efforts on results and problems dealing with spanning cycles (the classic hamiltonian problem) in ordinary graphs. I shall not attempt to survey digraphs, the traveling salesman problem (see instead [107]), or any of its related questions. However, I shall mention a few related results. I shall further restrict my attention primarily to work done since the late 1970s; however, for completeness, I shall include some earlier work in several places. For an excellent general introduction to the hamiltonian problem, the reader should see the article by