Counting odd cycles in locally dense graphs

Approximately Counting Hamilton Paths and Cycles in Dense Graphs

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (g + e)n, for any fixed e > 0. We show that the exact counting problems are #P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs.

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c > 3/8 contains (cn/e)(1 + o(1)) directed Hamilton cycles. This is an extension of a result of Cuckler, who settle...

Small odd cycles in 4-chromatic graphs

It is shown that every 4-chromatic graph on n vertices contains an odd cycle of length less than 2  p n‡3. This improves the previous bound given by Nilli [J Graph Theory 3 (1999), 145±147]. ß 2001 John Wiley & Sons, Inc. J Graph Theory 37: 115±117, 2001

On the Odd Cycles of Normal Graphs

A graph is normal if there exists a cross-intersecting pair of set families one of which consists of cliques while the other one consists of stable sets, and furthermore every vertex is obtained as one of these intersections. It is known that perfect graphs are normal while Cs, CT. and (_: are not. We conjecture that these three graphs are the only minimally not normal graphs. We give sufficien...

Enumerating and Counting Cycles in Bipartite Graphs