There have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K,,,-free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K,.3-free graphs.
A graph is locally connected if every neighborhood induces a connected subgraph. We show here that every connected, locally connected graph on p r 3 vertices and having no induced K,,3 is Hamiltonian. Several sufficient conditions for a line graph to be Hamiltonian are obtained as corollaries.
In this article w e show that the standard results concerning longest paths and cycles in graphs can be improved for K,,,-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K,,,-free graphs.There have been a great many results in recent years dealing with graphs that do not contain a copy of K l , , as an induced subgraph. It appears that this class of graphs is better behaved, in many respects, than graphs in general. It has been shown that for such graphs G, (i) If G is connected of even order then G has a I-factor Throughout this paper 6 will denote the minimal degree of a vertex in the graph. Other terminology not defined in this paper will agree with that in Behzad and Chartrand [2].For graphs that are not necessarily hamiltonian there are several results dealing wth longest paths and cycles. Dirac has shown [3] that if G is a connected graph then G has a path of length 26 or a hamiltonian path. Dirac also proved that if G is a 2-connected graph, then G has a cycle of length at least 26 or a hamiltonian cycle. The examples showing that Dirac's results are sharp are not K,,,-free, and if we add K,,3-free to the hypothesis, we obtain slightly larger lower bounds on the longest paths and cycles in graphs. To prove this, we first need a couple of lemmas about maximal cycles and paths in K,,,-free graphs. The proofs of the these lemmas are straightforward and hence are not included.
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