Vertex-disjoint Edges Research Articles

Thrackles, Superthrackles and the Hanani-Tutte Theorem

A thrackle is a drawing of a graph in which any two vertex-disjoint edges cross exactly once and incident edges do not cross. A graph that has a thrackle drawing is thracklable. In the past three decades, mathematicians investigated a number of variations of thrackles by relaxing or changing the required number of crossings on edges or restricting the placement of vertices on the surface (e.g., restricting to outerdrawings, in which vertices are restricted to a boundary curve of the surface). A graph is superthracklable if it can be drawn so that any two edges cross exactly once. We provide a simple proof for the following: a graph can be drawn on the plane so that every two edges cross an odd number of times if and only if it is superthracklable on the plane. A graph is outersuperthracklable if it has a drawing on a disc in which every vertex is on the boundary of the disc and every pair of edges cross exactly once. We introduce some natural generalisations of outersuperthracklable graphs and we show that these classes of graphs are equivalent. Lastly, using the Hanani-Tutte characterisation of planar graphs we show that for any surface $\Sigma$, there is a relationship between the class of graphs that are not embeddable on $\Sigma$ and the class of graphs that are not superthracklable with respect to $\Sigma$. More specifically, we show how to construct, from any forbidden minor $G$ for embeddability in $\Sigma$, two infinite families of graphs that are not superthracklable with respect to $\Sigma$. This sheds further light on the relationship between some of Archdeacon and Stor's forbidden configurations for superthrackles and Kuratowski's forbidden minors for planarity.

Solving maximum weighted matching on large graphs with deep reinforcement learning

Maximum weighted matching (MWM), which finds a subset of vertex-disjoint edges with maximum weight, is a fundamental topic with a wide spectrum of applications in various fields, such as biotechnology, social analysis, web services, etc. However, traditional methods cannot achieve a good balance between cost and quality. Inspired by the reinforcement learning techniques, we propose an MWM framework, L2M, with a Deep Reinforcement Learning (DRL) model, which can efficiently and effectively solve the MWM problem on large-scale general graphs. First, in contrast to traditional DRL methods that define the Markov Decision Process (MDP) without considering the efficiency issue, we represent MWM as an MDP that is carefully designed to accelerate computation. In particular, this MDP supports selecting multiple edges for each action and uses a pruning method to reduce search space efficiently. Second, since none of the existing DRL methods can support edge selection on large-scale graphs efficiently, we propose an edge-message-passing network EEN to generate edge embedding. To the best of our knowledge, this is the first attempt that uses DRL model for solving the MWM problem. Experimental results show that L2M outperforms state-of-the-art algorithms and generalizes well on graphs of different sizes and distributions.

Maximum density of vertex-induced perfect cycles and paths in the hypercube

Let H and K be subsets of the vertex set V(Qd) of the d-cube Qd (we call H and K configurations in Qd). We say K is an exact copy of H if there is an automorphism of Qd which sends H to K. If d is a positive integer and H is a configuration in Qd, we define λ(H,d) to be the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, and make conjectures about λ(C2d,d) and λ(Pd+1,d) for larger values of d. In our proofs there are connections with counting the number of sequences with certain properties and with the inducibility of certain small graphs. In particular, we needed to determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.

Strong and weighted matchings in inhomogenous random graphs

We equip the edges of a deterministic graph H with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in H satisfying the property that end-vertices of any two distinct edges are at least a minimum distance apart. We call such matchings as strong matchings and determine bounds on the expectation and variance of the minimum weight of a maximum strong matching. Next, we consider an inhomogenous random graph whose edge probabilities are not necessarily the same and determine bounds on the maximum size of a strong matching in terms of the averaged edge probability. We use local vertex neighbourhoods, the martingale difference method and iterative exploration techniques to obtain our desired estimates.

Fault-free Hamiltonian cycle including given edges in folded hypercubes with faulty edges

The folded hypercube is an important interconnection network for multiprocessor systems. Let [Formula: see text] with [Formula: see text] denote an [Formula: see text]-dimensional folded hypercube. For a given fault-free edge set [Formula: see text] with [Formula: see text] and a faulty edge set [Formula: see text] with [Formula: see text], in this paper we prove that [Formula: see text] contains a fault-free Hamiltonian cycle including each edge of [Formula: see text] if and only if the subgraph induced by [Formula: see text] is linear forest. Furthermore, we give the definitions of the distance among three vertex-disjoint edges and the distance between a vertex and a vertex set. For three vertex-disjoint edges [Formula: see text], the distance among them is denoted by [Formula: see text]. For a vertex [Formula: see text] and a vertex set [Formula: see text], the distance between [Formula: see text] and [Formula: see text] is denoted by [Formula: see text].

Number of 1-Factorizations of Regular High-Degree Graphs

A 1-factor in an n-vertex graph G is a collection of $$\frac{n}{2}$$ vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors. Clearly, a 1-factorization of G cannot exist unless n is even and G is regular (that is, all vertices are of the same degree). The problem of finding 1-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has a 1-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, Kuhn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950s, which says that every d-regular graph on n vertices contains a 1-factorization, provided that n is even and $$d \geqslant 2\left[ {\frac{n}{4}} \right] - 1$$ . In this paper we address the natural question of estimating F(n, d), the number of 1-factorizations in d-regular graphs on an even number of vertices, provided that $$d \geqslant \left[ {\frac{n}{2}} \right] + \varepsilon n$$ . Improving upon a recent result of Ferber and Jain, which itself improved upon a result of Cameron from the 1970s, we show that $$F\left( {n,\,d} \right) \geqslant {\left( {\left( {1 + o\left( 1 \right)} \right)\frac{d}{{{e^2}}}} \right)^{nd/2}}$$ , which is asymptotically best possible.

A Turán Theorem for Extensions Via an Erdős-Ko-Rado Theorem for Lagrangians

The extension of an r-uniform hypergraph G is obtained from it by adding for every pair of vertices of G, which is not covered by an edge in G, an extra edge containing this pair and (r−2) new vertices. In this paper we determine the Turan number of the extension of an r-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turan number for r=3. As the key ingredient of the proof we show that the Lagrangian of intersecting r-graphs is maximized by principally intersecting r-graphs for r≥4.

Doubly Resolvable Canonical Kirkman Packing Designs and its Applications

Let $$v \equiv 4 \pmod 6$$ and $$v\ge 4$$ . A canonical Kirkman packing design of order v, denoted by $$\hbox {CKPD}(v)$$ , is a resolvable packing with $$r=(v-4)/2$$ parallel classes such that (i) each parallel class consists of a size 4 block and $$(v- 4)/3$$ triples; (ii) the leave consists of the union of $$(v- 4)/2$$ vertex-disjoint edges and a $$K_4$$ with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable $$\hbox {CKPD}(v)\hbox {s}$$ with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained.

Polynomial size linear programs for problems in P

A perfect matching in an undirected graph G=(V,E) is a set of vertex disjoint edges from E that include all vertices in V. The perfect matching problem is to decide if G has such a matching. Recently Rothvoß proved the striking result that the Edmonds’ matching polytope has exponential extension complexity. In this paper for each n=|V| we describe a polytope for the perfect matching problem that is different from Edmonds’ polytope and define a weaker notion of extended formulation. We show that the new polytope has a weak extended formulation (WEF) Q of polynomial size. For each graph G with n vertices we can readily construct an objective function so that solving the resulting linear program over Q decides whether or not G has a perfect matching. With this construction, a straightforward O(n4) implementation of Edmonds’ matching algorithm using O(n2) bits of space would yield a WEF Q with O(n6logn) inequalities and variables. The construction is uniform in the sense that, for each n, a single polytope is defined for the class of all graphs with n nodes. The method extends to solve polynomial time optimization problems, such as the weighted matching problem. In this case a logarithmic (in the weight of the optimum solution) number of optimizations are made over the constructed WEF.The method described in the paper involves the construction of a compiler that converts an algorithm given in a prescribed pseudocode into a polytope. It can therefore be used to construct a polytope for any decision problem in P which can be solved by a well defined algorithm. Compared with earlier results of Dobkin–Lipton–Reiss and Valiant our method allows the construction of explicit linear programs directly from algorithms written for a standard register model, without intermediate transformations. We apply our results to obtain polynomial upper bounds on the non-negative rank of certain slack matrices related to membership testing of languages in P/poly.

Multidecomposition of cartesian product of some graphs into even cycles and matchings

Let C2p and pK2 denote a cycle with 2p edges and p vertex-disjoint edges, respectively. For graphs G,H' and H'',a (H',H'')-multidecomposition of G is a partition of the edge set of G into copies of H' and copies of H'' with at least one copy of H' and at least one copy of H''. In this paper, we investigate (C2p, pK2)-multidecomposition of the Cartesian product of paths, cycles and complete graphs, for some values p ? 3.

Reconstruction of the Geometric Structure of a Set of Points in the Plane from Its Geometric Tree Graph

Let P be a finite set of points in general position in the plane. The structure of the complete graph K(P) as a geometric graph includes, for any pair [a, b], [c, d] of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of K(P) are the vertices of the so-called Geometric Tree Graph of P, G(P). Two such vertices are adjacent in G(P) if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. In this paper we show how to reconstruct from G(P) (regarded as an abstract graph) the structure of K(P) as a geometric graph. We first identify within G(P) the vertices that correspond to spanning stars. Then we regard each star S(z) with center z as the representative in G(P) of the vertex z of K(P). (This correspondence is determined only up to an automorphism of K(P) as a geometric graph.) Finally we determine for any four distinct stars S(a), S(b), S(c), and S(d), by looking at their relative positions in G(P), whether the corresponding segments cross.

Matching critical intersection hypergraphs

A matching in a hypergraph H is a set of pairwise vertex disjoint edges in H and the matching number of H is the maximum cardinality of a matching in H. A hypergraph H is an intersecting hypergraph if every two distinct edges of H have a non-empty intersection. Equivalently, H is an intersecting hypergraph if and only if it has matching number one. In this paper, we study intersecting hypergraphs that are matching critical in the sense that the matching number increases under various definitions of criticality.

An improved approximation ratio for the jump number problem on interval orders

The jump number problem is to find a linear extension of a given poset that minimizes the number of incomparable adjacent elements. The problem is NP-hard even in the class of interval orders and so three different 3/2-approximation algorithms for this case have been proposed respectively by Felsner, Sysło, and Mitas. We build upon the approach of Mitas, where the problem is reduced to packing vertex-disjoint edges and odd cycles in a certain graph. We prove that the requirement of independence between edges and cycles may be abandoned. In effect, the jump number problem is reduced to the set cover problem, and the 4/3-approximation algorithm of Duh and Fürer for the 3-set cover problem is used to show that the approximation ratio of the jump number problem on interval orders is tighter than 89/60.

Forbidden subgraphs and the König–Egerváry property

The matching number of a graph is the maximum size of a set of vertex-disjoint edges. The transversal number is the minimum number of vertices needed to meet every edge. A graph has the König–Egerváry property if its matching number equals its transversal number. Lovász proved a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs within graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovász’s result to a characterization of all graphs having the König–Egerváry property in terms of forbidden configurations (which are certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. Using our characterization of graphs with the König–Egerváry property, we also prove a forbidden subgraph characterization for the class of edge-perfect graphs.

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m 2 / e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.

Maximum Induced Matchings for Chordal Graphs in Linear Time

The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G *=L(G)2 of the line graph L(G) of G, and in some cases, G * is in the same graph class; for example, for chordal graphs G, G * is chordal. The construction of G *, however, requires ${\mathcal{O}}(m^{2})$time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.

The existence of restricted resolvable designs I: (1,2)-factorizations of K2 n

It has been known for at least a century that (in modern terminology) the complete graph K 2 n admits a 1-factorization, that is, a partition of its edge set E into subsets E 1,…, E 2 n−1 such that each E i consists of n vertex-disjoint edges. A considerably newer result (due to Ray-Chaudhuri and Wilson) states that if n is an odd integer then the complete graph K 3 n admits what we will call a 2-factorization, that being a pair ( T, P) where T is a decomposition of K 3 n into triangles ( K 3's) and P is a partition of T into subsets T 1,…, T (3n−1) 2 so that each T i consists of n vertex-disjoint triangles. Between these two extremes we define a (1, 2)- factorization of K n with cardinality k to be a pair ( T, P) where T is a decomposition of K n into edges and triangles ( K 2's and K 3's) and P is a partition of T into subsets T 1,…, T k such that each T i is a vertex partition of K n . This is the first in a series of two papers in which we investigate the following question: for which integers n > 0 and ⌊ n/2⌋ ⩽ k ⩽ n − 1 does the complete graph K n admit a (1, 2)-factorization of cardinality k? We prove here that when n is even the ‘obvious’ necessary conditions for the existence of these designs are sufficient, with exactly two exceptions: n = 6, k = 3; and n = 12, k = 6.

On certain classes of fractional matchings

An f-matching in an undirected graph X is defined as a set of vertex disjoint edges and odd cycles. In particular we consider f-matchings which saturate the maximum possible number of vertices and contain a maximum number of vertex disjoint edges. The main result is, that in this case different possible f-matchings in X with these properties contain the same number of triangles, pentagons and so on. This means that maximizing the set of vertex disjoint edges in the f-matchings determines the number of cycles of length 3 (i.e. triangles), 5,…,(2 n + 1). The problem is stated as a linear programming problem called fractional matching problem in a graph X.