Related papers: A graph theoretic proof of Cramer's rule

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory,…

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…

We extend the closed graph theorem and the open mapping theorem to a context in which a natural duality interchanges their extensions.

An technically interesting proof of a known theorem.

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A…

This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications.

Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich…

The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road…

We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function…

We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley--Stembridge conjecture.

We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lov\'asz, Catlin, Kostochka and Rabern.

In this paper, we give a proof of the Bouchard-Klemm-Marino-Pasquetti conjecture for a framed vertex, by using the symmetrized Cut-Join Equation developed in a previous paper.

We introduce a new method for decomposing the edge set of a graph, and use it to replace the Regularity lemma of Szemer\'edi in some graph embedding problems. An algorithmic version is also given.

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to…

The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated…

A new very simple proof of the number of labeled rooted forest-graphs with a given number of vertices is given. As a partial case of this formula we have Cayley's formula.

The Laplacian matrix $L$ of a signed graph $G$ may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of $L$. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an…

Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G (giving at each m>0 the…

We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality…

For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings.…