Perfect graphs of fixed density: counting and homogenous sets

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomass\'e [Almost all H-free graphs have the Erd\H{o}s-Hajnal property] in the case of forbidden induced C_5.Comment: 19 page

Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.Comment: 14 page

On the Chromatic Thresholds of Hypergraphs

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure