Related papers: Generalized Whitney formulas for broken circuits i…

Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…

We establish two expansions of the Potts model partition function of a graph. One is along the deletions of a graph, a rewritten formula given in Biggs (1977). The other is along the contractions of a graph. Then, we specialize the…

We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie…

In this article we consider certain well-known polynomials associated with graphs including the independence polynomial and the chromatic polynomial. These polynomials count certain objects in graphs: independent sets in the case of the…

In a recent paper by M.Kazarian and the second author, a recurrence for the Lie algebras $\mathfrak{so}(N)$ weight systems has been suggested; the recurrence allows one to construct the universal $\mathfrak{so}$ weight system. The…

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated…

Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

In this paper we give a new characterization of the h-vector of the chromatic polynomial of a graph. We introduce reduced chromatic cohomology of a graph and show that h_i are its Betti numbers. We then discuss various combinatorial…

Given a graph $G$ of order $n$, the $\sigma$-$polynomial$ of $G$ is the generating function $\sigma(G,x) = \sum a_{i}x^{i}$ where $a_{i}$ is the number of partitions of the vertex set of $G$ into $i$ nonempty independent sets. Such…

The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic…

The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for…

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known…

Brown has shown that the Stanley-Reisner ring of the broken circuit complex of a graph has a linear system of parameters which is defined in terms of the circuits and cocircuits of the graph. Later on Brown and Sagan conjectured a special…

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very…

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$.…

The central open question of algebraic complexity is whether VP is unequal to VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been…

We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $\chi_G(k,l)$ that counts the number of signed colorings using colors…

Let $T$ be a forest. We study polynomially high-chromatic pure pairs in graphs with no $T$ as an induced subgraph ($T$-free graphs in other words), with applications to the polynomial Gy\'arf\'as-Sumner conjecture. In addition to reproving…