Random perfect matchings in regular graphs

We prove that in all regular robust expanders $G$, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.     

Skew partitions in perfect graphs

We discuss some new and old results about skew partitions in perfect graphs.     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . 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 and clique number 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 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 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 .     

Proof of a conjecture on irredundance perfect graphs

Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H)=γ(H), for every induced subgraph H of G. In this article we present a result which immediately implies three known conjectures on irredundance perfect graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 292–306, 2002     

Forbidden induced subgraph characterizations of subclasses and variations of perfect graphs: A survey

A graph is perfect if the chromatic number is equal to the clique number for every induced subgraph of the graph. Perfect graphs were defined by Berge in the sixties. In this survey we present known results about partial characterizations by forbidden induced subgraphs of different graph classes related to perfect graphs. We analyze a variation of perfect graphs, clique-perfect graphs, and two subclasses of perfect graphs, coordinated graphs and balanced graphs.     

New conjectures on perfect matchings in cubic graphs

We propose three new conjectures on perfect matchings in cubic graphs. The weakest conjecture is implied by a well-known conjecture of Berge and Fulkerson. The other two conjectures are a strengthening of the first one. All conjectures are trivially verified for 3-edge-colorable cubic graphs and by computer for all snarks of order at most 34.     

On perfect matchings in matching covered graphs

A graph is matching-covered if every edge of is contained in a perfect matching. A matching-covered graph is strongly coverable if, for any edge of , the subgraph is still matching-covered. An edge subset of a matching-covered graph is feasible if there exist two perfect matchings and such that , and an edge subset with at least two edges is an equivalent set if a perfect matching of contains either all edges in or none of them. A strongly matchable graph does not have an equivalent set, and any two independent edges of form a feasible set. In this paper, we show that for every integer , there exist infinitely many -regular graphs of class 1 with an arbitrarily large equivalent set that is not switching-equivalent to either or , which provides a negative answer to a problem of Lukot’ka and Rollová. For a matching-covered bipartite graph , we show that has an equivalent set if and only if it has a 2-edge-cut that separates into two balanced subgraphs, and is strongly coverable if and only if every edge-cut separating into two balanced subgraphs and satisfies and .     

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

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, we obtain a polynomial delay algorithm for listing the anti‐vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d‐factors in bipartite graphs, and perfect 2‐matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006     

Claw-free graphs with strongly perfect complements. Fractional and integral version. Part I. Basic graphs

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Norms and perfect graphs

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods.In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium on Operations Research, Passau, 9.–11. 9. 1987.     

On the perfect orderability of unions of two graphs

A graph G is perfectly orderable, if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph, if it contains no P₄, 2K₂, and C₄ as induced subgraph. A theorem of Chvátal, Hoàng, Mahadev, and de Werra states that a graph is perfectly orderable, if it is the union of two threshold graphs. In this article, we investigate possible generalizations of the above theorem. Hoàng has conjectured that, if G is the union of two graphs G₁ and G₂, then G is perfectly orderable whenever G₁ and G₂ are both P₄‐free and 2K₂‐free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter‐example to this conjecture, and we prove a special case of it: if G₁ and G₂ are two edge‐disjoint graphs that are P₄‐free and 2K₂‐free, then the union of G₁ and G₂ is perfectly orderable. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 32–43, 2000     

Claw-free graphs with strongly perfect complements. Fractional and integral version, Part II: Nontrivial strip-structures

Strongly perfect graphs have been studied by several authors (e.g., Berge and Duchet (1984) [1], Ravindra (1984) [7] and Wang (2006) [8]). In a series of two papers, the current paper being the second one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [8] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.     

On a connection between facility location and perfect graphs

We characterize the graphs for which a linear relaxation of a facility location problem defines a polytope with all integral extreme points. We use a transformation to a stable set problem in perfect graphs. Based on this transformation, these graphs can be recognized in polynomial time.     

Graphs with independent perfect matchings

A graph with at least two vertices is matching covered if it is connected and each edge lies in some perfect matching. A matching covered graph G is extremal if the number of perfect matchings of G is equal to the dimension of the lattice spanned by the set of incidence vectors of perfect matchings of G. We first establish several basic properties of extremal matching covered graphs. In particular, we show that every extremal brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels. Then, using the main theorem proved in 2 and 3 , we find all the extremal cubic matching covered graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 19–50, 2005