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 Chvátal and Erdős. 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 α(G) is the maximum integer r such that G contains an (s, t )-bipartite-hole for every pair of nonnegative integers s and t with s + t = r. Our central theorem is that a graph G with at least three vertices is Hamiltonian if its minimum degree is at least α(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 α(G). C 2017 Wiley Periodicals, Inc.

We investigate the asymptotic structure of a random perfect graph Pn sampled uniformly from the set of perfect graphs on vertex set {1,…,n}. Our approach is based on the result of Prömel 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 α(Pn) and clique number ω(Pn) is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that Pn contains any given graph H as an induced subgraph is asymptotically 0 or 12 or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity κ(Pn) equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon WP(x,y)=12 (1[x≤1/2]+ 1[y≤1/2]).

A graph is unipolar if it can be partitioned into a clique and a disjoint union of cliques, and a graph is a generalised split graph if it or its complement is unipolar. A unipolar partition of a graph can be used to find efficiently the clique number, the stability number, the chromatic number, and to solve other problems that are hard for general graphs. We present an O(n 2 )-time algorithm for recognition of n-vertex generalised split graphs, improving on previous O(n 3 )-time algorithms.

In this paper we present a new width measure for a tree decomposition, minor-matching hypertree width, µ-tw, for graphs and hypergraphs, such that bounding the width guarantees that set of maximal independent sets has a polynomially-sized restriction to each decomposition bag. The relaxed conditions of the decomposition allow a much wider class of graphs and hypergraphs to have bounded width compared to other tree decompositions. We show that, for fixed k, there are 2n-vertex graphs of minor-matching hypertree width at most k. A number of problems including Maximum Independence Set, k-Colouring, and Homomorphism of uniform hypergraphs permit polynomial-time solutions for hypergraphs with bounded minor-matching hypertree width and bounded rank. We show that for any given k and any graph G, it is possible to construct a decomposition of minor-matching hypertree width at most O(k 3 ), or to prove that µ-tw(G) > k in time nThis is done by presenting a general algorithm for approximating the hypertree width of well-behaved measures, and reducing µ-tw to such measure. The result relating the restriction of the maximal independent sets to a set S with the set of induced matchings intersecting S in graphs, and minor matchings intersecting S in hypergraphs, might be of independent interest.

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