A set of vertices in a graph is perfect dominating if every vertex outside the set is adjacent to exactly one vertex in the set, and is neighborhood connected if the subgraph induced by its open neighborhood is connected. In any graph the full set of vertices is perfect dominating, and in every connected graph the full set of vertices is neighborhood connected. It is shown that (i) in a connect...

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). This result relies on polyhedral characterizations of perfect graphs involving the stable set polytope of the graph, a linear relaxation defined by clique constraints, and a semi-definite relaxation, the Theta-body of the gr...

We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1, x] satisfies estimates of the form exp((log x) log log log )...

A (proper) coloring of a finite simple graph (G) is pe#ect if it uses exactly o(G) colors, where o(G) denotes the order of a largest clique in G. A coloring is locally-perfect [3] if it induces on the neighborhood of every vertex v a perfect coloring of this neighborhood. A graph G is perfect (resp. locally-petfect) if every induced subgraph admits a perfect (resp. locally-perfect) coloring. Pr...

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the payoff sets are typically fractals unless they are full-dimensional, which may happen when the discount factors are large enough. More specifically, the equilibrium payoffs can be identified as subsets of self-affine sets or graph-directed self-affine ...

We refer the reader to Jerrum's book [1] for the analysis of a Markov chain for generating a random matching of an arbitrary graph. Here we'll look at how to extend the argument to sample perfect matchings in dense graphs and arbitrary bipartite graphs. Both of these results are due to Jerrum and Sinclair [2]. Dense graphs We'll need some definitions and notations first: Definition 7.1 A graph ...

In 1961, Claude Berge proposed the \strong perfect graph conjecture", probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview of the solution, together with an account of some of the ideas that eventually brought us to the answer.

A graph G is perfect if, for all induced subgraphs of G, the size of a largest clique is equal to the chromatic number. A graph is minimally imperfect if it is not perfect but all its proper induced subgraphs are. A hole is a chordless cycle of length at least four. The strong perfect graph conjecture of Berge [1] states that G is minimally imperfect if and only if G or its complement is an odd...

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect Hpackings: Given n, r,D ∈ N, we characterise the edge density threshold that ensures a perfect Kr-packing in any graph G on n vertices and with minimum degree δ(G) ≥ D. We also give two conjectures concerning degree...

In order to obtain perfect state transfer between two sites in a network of interacting qubits, their corresponding vertices in the underlying graph must satisfy a property called strong cospectrality. Here we determine the structure of graphs containing pairs of vertices which are strongly cospectral and satisfy a certain extremal property related to the spectrum of the graph. If the graph sat...