A graph G is 1-extendable or almost 1-extendable if every edge is contained in a perfect or almost perfect matching of G, respectively. Let d ≥ 3 be an integer, and let G be a graph of order n with exactly one odd component such that the degree of each vertex is either d or d + 1. If G is not almost 1-extendable, then we prove that n ≥ 2d + 5. In the special case that d ≥ 4 is even and G is a d...

Using elements of the structural theory of matchings and a recently proved conjecture concerning bricks, it is shown that every n-extendable brick (except K4, C6 and the Petersen graph) with p vertices and q edges contains at least q − p + (n − 1)!! perfect matchings. If the girth of such an n-extendable brick is at least five, then this graph has at least q − p + nn−1 perfect matchings. As a c...

Let us improve this bound. Assume that G is a connected graph and T is its spanning tree rooted at r. Let us consider an ordering of V (G) in which each vertex v appears after its children in T . Now, for v 6= r we have |N(vi) ∩ {v1, . . . , vi−1}| ≤ deg v − 1, so c(vi) ≤ deg vi for vi 6= r. Unfortunately, the greedy may still need to use ∆(G) + 1 colors if deg r = ∆(G) and each child of r happ...

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite...

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, this gives a polynomial delay algorithm for listing the...

While the famous Berge’s Strong Perfect Graph Conjecture (see [l] for details on perfect graphs) remains a major unsolved problem in Graph Theory, an alternative characterization of Perfect Graphs was conjectured in 1982 by Berge and the author [3]. This second conjecture asserts the existence of kernels for a certain type of orientations of perfect graphs. Here we prove a weaker form of the co...

The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent if their symmetric difference forms the edge set of a hexagon of B . A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B−P contains at least one perfect matching, or if B − P is empty. It is proven that there exists a surjective map ...

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite...

the idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as latin squares, block designs and steiner systems in combinatorics (see [1] and the references therein). recently, the forcing on perfect matchings has been attracting more researchers attention. a forcing set of m is a subset of m contained...

In this paper we prove that perfect graphs are kernel solvable, as it was conjectured by Berge and Duchet (1983). The converse statement, i.e. that kernel solvable graphs are perfect, was also conjectured in the same paper, and is still open. In this direction we prove that it is always possible to substitute some of the vertices of a non-perfect graph by cliques so that the resulting graph is ...