A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|). This is independent of the recent proof of the strong perfect graph conjecture.

We show that deciding whether a given graph G of size m has a unique perfect matching as well as finding that matching, if it exists, can be done in time O(m) if G is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.

This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then  (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. We characterise bipartite graphs and near-bipartite graphs whose perfect matching polytopes have diameter 1.

We prove that a 2K 2 minimal imperfect Berge graph (if any) contains a K 6 : A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by (H)) not exceeding the cardinality !(H) of a maximum clique of H. A graph is minimal imperfect if all its proper induced subgraphs are perf...

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K3,3-minor-free graphs by building on Kasteleyn’s scheme for planar graphs, and stated that this “opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K3,3-free graphs.” In this paper, we finally settle this 30-year-old open problem. Building on the recent breakthrough resu...

A kernel of a directed graph D is defined as an independent set which is reachable from each outside vertex by an arc. A graph G is called kernel-solvable if an orientation D of G has a kernel whenever each clique of G has a kernel in D. The notion of kernel-solvability has important applications in combinatorics, list coloring, and game theory. It turns out that kernel-solvability is equivalen...

We will characterize all graphs that have the property that the graph and its complement are minimal even pair free. This characterization allows a new formulation of the Strong Perfect Graph Conjecture. The reader is assumed to be familiar with perfect graphs (see for example [2]). A hole is a cycle of length at least five. An odd hole is a hole that has an odd number of vertices. An (odd) ant...

We are interested in a version of graph colouring where there is a ‘co-site’ constraint value k. Given a graph G with a non-negative integral demand xv at each node v, we must assign xv positive integers (colours) to each node v such that the same integer is never assigned to adjacent nodes, and two distinct integers assigned to a single node differ by at least k. The aim is to minimise the spa...

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. The seminal Hajnal–Szemerédi theorem [12] characterises the minimum degree that ensures a graph G contains a perfect Kr-packing. Balogh, Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal–Szemerédi theorem which, if true, gives a strength...