Random Perfect Graphs

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $\alpha(P_n)$ and clique number $\omega(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any given graph $H$ as an induced subgraph is asymptotically $0$ or $\frac12$ or $1$. Further we show that almost all perfect graphs are $2$-clique-colourable, improving a result of Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have connectivity $\kappa(P_n)$ equal to their minimum degree; they are almost all in class one (edge-colourable using $\Delta$ colours, where $\Delta$ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon $W_P(x, y) = \frac12(\mathbb{1}[x \le 1/2] + \mathbb{1}[y \le 1/2])$

Hamilton cycles, minimum degree and bipartite holes

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chv\'atal and Erd\H{o}s. In detail, an $(s, t)$-bipartite-hole in a graph $G$ consists of two disjoint sets of vertices $S$ and $T$ with $|S|= s$ and $|T|=t$ such that there are no edges between $S$ and $T$; and $\widetilde{\alpha}(G)$ is the maximum integer $r$ such that $G$ contains an $(s, t)$-bipartite-hole for every pair of non-negative integers $s$ and $t$ with $s + t = r$. Our central theorem is that a graph $G$ with at least $3$ vertices is Hamiltonian if its minimum degree is at least $\widetilde{\alpha}(G)$. From the proof we obtain a polynomial time algorithm that either finds a Hamilton cycle or a large bipartite hole. The theorem also yields a condition for the existence of $k$ edge-disjoint Hamilton cycles. We see that for dense random graphs $G(n,p)$, the probability of failing to contain many edge-disjoint Hamilton cycles is $(1 - p)^{(1 + o(1))n}$. Finally, we discuss the complexity of calculating and approximating $\widetilde{\alpha}(G)$

Homogeneous sets in graphs and hypergraphs

A set of vertices in a graph or a hypergraph is called homogeneous if it is independent, that is it does not contain any edge, or if it is complete, that is it contains all possible pairs or subsets of it as edges. We investigate the properties of graphs and hypergraphs in two cases of imposed restrictions on the structure of their homogeneous sets. First we study the asymptotic structure of random perfect graphs. We give a generation model which yields such graphs almost uniformly, with an additive error of e-Ω(n) in the total variation distance. We use this model to determine a number of properties of random perfect graphs, including the distribution of the stability and the clique number, the probability of containing a fixed induced subgraph, Hamiltonicity, clique-colourability, connectivity, edge colouring, and the limit of a uniformly drawn sequence of perfect graphs. In the second part, we give a hypergraph parameter μ(H), called minor- matching number, with the property that hypergraphs H with bounded rank and minor-matching number contain a polynomially-bounded number of maximal independent sets. In the other direction, every hypergraph H contains at least 2μ(H) maximal independent sets. A number of hard hypergraph problems, including maximum-sized independent set, k-colouring and hypergraph homomorphism can be solved in polynomial time if a list with all maximal independent sets of the hypergraph is given as part of the input, and hence a family of instances with bounded minor-matching number of the input hypergraph form a new polynomial class for the problems above. The class can further be generalised by considering the maximum minor matching number of a bag in a tree decomposition as a new treewidth measure. We explain how to use this measure, defined as minor-matching treewidth, to solve hard problems and how to algorithmically construct a tree decomposition with approximate minimal width.</p

Homogeneous sets in graphs and hypergraphs

A set of vertices in a graph or a hypergraph is called homogeneous if it is independent, that is it does not contain any edge, or if it is complete, that is it contains all possible pairs or subsets of it as edges. We investigate the properties of graphs and hypergraphs in two cases of imposed restrictions on the structure of their homogeneous sets. First we study the asymptotic structure of random perfect graphs. We give a generation model which yields such graphs almost uniformly, with an additive error of e-Ω(n) in the total variation distance. We use this model to determine a number of properties of random perfect graphs, including the distribution of the stability and the clique number, the probability of containing a fixed induced subgraph, Hamiltonicity, clique-colourability, connectivity, edge colouring, and the limit of a uniformly drawn sequence of perfect graphs. In the second part, we give a hypergraph parameter μ(H), called minor- matching number, with the property that hypergraphs H with bounded rank and minor-matching number contain a polynomially-bounded number of maximal independent sets. In the other direction, every hypergraph H contains at least 2μ(H) maximal independent sets. A number of hard hypergraph problems, including maximum-sized independent set, k-colouring and hypergraph homomorphism can be solved in polynomial time if a list with all maximal independent sets of the hypergraph is given as part of the input, and hence a family of instances with bounded minor-matching number of the input hypergraph form a new polynomial class for the problems above. The class can further be generalised by considering the maximum minor matching number of a bag in a tree decomposition as a new treewidth measure. We explain how to use this measure, defined as minor-matching treewidth, to solve hard problems and how to algorithmically construct a tree decomposition with approximate minimal width.</p