Related papers: Counting Perfect Matchings in Dense Graphs Is Hard
We prove that deciding whether a given input word contains as subsequence every possible permutation of integers $\{1,2,\ldots,n\}$ is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as…
Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The…
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
IV-matching is a generalization of perfect bipartite matching. The complexity of finding IV-matching in a graph was posted as an open problem at the ICALP 2014 conference. In this note, we resolve the question and prove that, contrary to…
Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing.…
In this paper, motivated by a question posed in \cite{AH}, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove…
We introduce a new -as far as we know- problem, according to which we are asked to match sequences of two digits in matrices having entries among those two digits (but others too) and prove that this problem is NP-complete
We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a…
We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.
We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…
The distinguishing result of this paper is a $\mathbf{P}$-time enumerable partition of all the potential perfect matchings in a bipartite graph. This partition is a set of equivalence classes induced by the missing edges in the potential…
In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent…
We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…
We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs. The model is based on the "demand query" model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching…
The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We…
We consider random perfect matchings on a general class of contracting bipartite graphs by letting certain edge weights be 0 on the contracting square-hexagon lattice in a periodic way. We obtain a deterministic limit shape in the scaling…
We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is…
In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the…
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite…