A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least 5 or the complement of one. The "strong perfect graph conjecture" (Berge, 1961) asserts that a graph is perfect if and only if it is Berge.
A stronger conjecture was made recently by Conforti, Cornuejols and Vuskovic -- that every Berge graph either falls into one of a few basic classes, or it has a kind of separation that cannot occur in a minimal imperfect graph.
In this paper we prove both these conjectures.
Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.
A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynom…
In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph.
A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is an extension of the famous Erdős-Ko-Rado (EKR) theorem \cite{EKR} to 2-intersecting families of perfect matchings for all values of $k$. Specifically, for $k\geq 3$ a set of 2-intersecting perfect matchings in $K_{2k}$ of maximum size has $(2k-5)(2k-7)\cdots (1)$ perfect matchings.
Two perfect matchings $P$ and $Q$ of the complete graph on $2k$ vertices are said to be set-wise $t$-intersecting if there exist edges $P_{1}, \cdots, P_{t}$ in $P$ and $Q_{1}, \cdots, Q_{t}$ in $Q$ such that the union of edges $P_{1}, \cdots, P_{t}$ has the same set of vertices as the union of $Q_{1}, \cdots, Q_{t}$ has. In this paper we prove an extension of the famous Erdős-Ko-Rado (EKR) theorem to set-wise $2$-intersecting families of perfect matching on all values of $k$, and we conjecture similar statement for all $t\geq 2$.
We study the perfect phylogeny problem and establish a generalization of the four gamete condition (also called the Splits Equivalence Theorem) for sequences over three state characters. Our main result is that a set of input sequences over three state characters allows a perfect phylogeny if and only if every subset of three characters allows a perfect phylogeny. In establishing these results, we prove fundamental structural features of the perfect phylogeny problem on three state characters and completely characterize the minimal obstruction sets that must occur in three state input sequence…
We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let δ_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} δ_{\rm Ore} (H,n)/n.
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 characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. Balogh, Kostochka and Treglown proposed a degree sequence version of the Hajnal--Szemerédi theorem which, if true, gives a strengthening of the Hajnal--Szemerédi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon--Yuster theorem which gives a minimum degree condit…
A $2k$-matching is a perfect matching of the complete graph on $2k$ vertices. Two $2k$-matchings are defined to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is that if $k \geq 3t/2+1$, then the largest system of $t$-intersecting $2k$-matchings has size $(2(k-t)-1)!! = \prod_{i=0}^{k-t-1}(2k-2t-2i-1)$ and the only systems that meet this bound consist of all $2k$-matchings that contain a set of $t$ disjoint edges. Further, this bound on $k$ is sharp for $t\geq 6$. The method used is this paper is similar to the proof of the complete Erdős-Ko-Rado t…
We correct the statements of two theorems and two corollaries in our paper [On subgroup perfect codes in Cayley graphs, European J. Combin. 91 (2021) 103228]. Proofs of these theorems and three other results are given as well.
This paper proves a new general K-network constrained energy reliability global factorization theorem. As in the unconstrained case, beside its theoretical mathematical importance the theorem shows how to do parallel processing in exact network constrained energy reliability calculations in order to reduce the processing time of this NP-hard problem. Followed by a new simple factorization formula for its calculation, we propose a new definition of constrained energy network reliability motivated by the factorization theorem and the accomplishment of parallel processing, something impossible wi…
In this paper we enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. The weak modular product differs from the direct product by also encoding non-adjacencies of the factor graphs in its edges. This work is motivated by the following: a 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Furthermore, a straightforward corollary of Kozen's theorem and Lovász's sandwich theorem is if the weak modular product betwee…
A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching widt…
Win [\emph{J. Graph Theory} {\bf 6}(1982), 489--492] conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\geq k+2$. In this paper, we prove that Win's conjecture is true for $k\geq n/2$, where $n$ is sufficiently large. To show this result, we prove a theorem on $k$-factor in a graph under some Ore-type condition. Our main tools include Tutte's $k$-factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization, and the solution to the longstanding 1-factor dec…
A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for exampl…
We consider the following generalization of the seminal Erdős-Ko-Rado theorem, due to Frankl. For k>= 2, let F be a k-wise intersecting family of r-subsets of an n element set X, i.e. any k sets in F have a nonempty intersection. If r<= (k-1/k)n, then |F|<={n-1 \choose r-1}. We extend Frankl's theorem in a graph-theoretic direction. For a graph G, and r>=1, let P^r(G) be the family of all r-subsets of the vertex set of G such that every r-subset is either an independent set or contains a maximum independent set. We will consider k-wise intersecting subfamilies of this family for the graph M_n,…
A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemerédi theorem determines the minimum degree threshold for forcing a perfect $K_r$-tiling in a graph $G$. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph $G$ labels from $\{-1,1\}$, and one seeks substructures $F$ of $G$ that have `high' discrepancy (i.e. the sum of the labels of the edges in $F$ is far from $0$). In this paper we determine the minimum degree thres…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide several new minimal non-strongly-perfect graphs.
A graph $G$ with a perfect matching is called saturated if $G+e$ has more perfect matchings than $G$ for any edge $e$ that is not in $G$. Lovász gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which revealed canonical structures of general graphs with perfect matchings. Here, the cathedral theorem is derived in quite a natural way, providing more refined or generalized properties. Moreover, the ne…
A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing, a tanglegram is called planar. We show that every non-planar tanglegram contains one of two non-planar 4-leaf tanglegrams as induced subtanglegram, which parallels Kuratowski's Theorem.
A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemerédi states that if $G$ is a graph of order $n$ with minimum degree $δ(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + γ)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore…
This paper has been withdrawn by the author. Peterson and Woodall previously proved that the list-edge-colouring conjecture holds for graphs without odd cycles of length 5 or longer. D. Peterson and D. R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Mathematics 202 (1999), 191-199. D. Peterson and D. R. Woodall, Erratum to "Edge-choosability in line-perfect multigraphs", Discrete Mathematics 260 (2003), 323-326.
Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Claude Berge made a conjecture about them, that was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and is now called the strong perfect graph theorem. This is a survey about perfect graphs, mostly focused on the strong perfect graph theorem.
We say that a (di)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 characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic …
Kang et al. provided a path realization of the crystal graph of a highest weight module over a quantum affine algebra, as certain semi-infinite tensor products of a single perfect crystal. In this paper, this result is generalized to give a realization of the tensor product of several highest weight modules. The underlying building blocks of the paths are finite tensor products of several perfect crystals. The motivation for this work is an interpretation of fermionic formulas, which arise from the combinatorics of Bethe Ansatz studies of solvable lattice models, as branching functions of affi…
We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $ω$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $ω$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least $3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to ea…
The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.
What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G …
A graph is strongly perfect if every induced subgraph H has a stable set that meets every maximal clique of H. A graph is claw-free if no vertex has three pairwise non-adjacent neighbors. The characterization of claw-free graphs that are strongly perfect by a set of forbidden induced subgraphs was conjectured by Ravindra in 1990 and was proved by Wang in 2006. Here we give a shorter proof of this characterization.
A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called {\em basic}. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by Lovász. This result is now known as the perfect graph theorem. …
A new shortest proof of Kotzig's Theorem about graphs with unique perfect matching is presented in this paper. It is well known that Kotzig's theorem is a consequence of Yeo's Theorem about edge-colored graph without alternating cycle. We present a proof of Yeo's Theorem based on the same ideas as our proof of Kotzig's theorem.
The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov's theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either…
A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize half-trees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian half-tree theorem, we obtain a refined version of the matrix-tree theorem in the case of skew-symmetric matrices, as wel…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of Kotzig's canonical partition to a decomposition of general graphs with perfect matchings. It contains a short proof for the theorem of the canonical partition. These results give decompositions which are canonical, that is, unique to given graphs. We also show that there are correlations between these two and that these can be computed in polynomial time.
Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdur…
