Related papers: Theorems, Problems and Conjectures
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special…
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as…
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 investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We…
Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and…
Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated…
A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a…
In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.
Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…
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…
In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously…
In this paper, two recursion formulae of chromatic polynomial of a maximal planar graph G are obtained. Moreover, the application of these formulaes to the proof of Four-Color Conjecture is investigated. By using these formulae, the proof…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of conjugate partitions. While there are exponentially many diagrams, we show it is…
For a simple graph $G$, let $\chi(G,x)$ denote the chromatic polynomial of $G$. This manuscript introduces some polynomials which are related to chromatic polynomial and their relations.
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…
We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…
We give an overview of recent progress around a problem introduced by Elekes and R\'onyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$,…