Related papers: Theorems, Problems and Conjectures
We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual…
In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials,…
In this paper, for each graph $G$, we def\mbox{}ine a chain complex of graded modules over the ring of polynomials, whose graded Euler characteristic is equal to the chromatic polynomial of $G$. Furthermore, we def\mbox{}ine a chain complex…
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by…
In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class,…
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…
We investigate the $\alpha$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the…
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that…
In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain…
This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same $U$-polynomial (or, equivalently, the same chromatic symmetric function). We consider the $U_k$-polynomial, which is a…
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct…
We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such…
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than…
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge…
Group field theories are higher dimensional generalizations of matrix models. Their Feynman graphs are fat and in addition to vertices, edges and faces, they also contain higher dimensional cells, called bubbles. In this paper, we propose a…