Related papers: $q$-Chromatic polynomials

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is…

We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric…

We prove necessary conditions for certain elementary symmetric functions, $e_\lambda$, to appear with nonzero coefficient in Stanley's chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do…

The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a…

The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$…

We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using…

A proper $k$-coloring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$, for every $uv\in E(G)$. The chromatic number $\chi(G)$ is the minimum $k$ such that there exists a proper $k$-coloring of $G$.…

For a graph $G$ and an integer $d\geq 0$, the defective chromatic polynomial $\chi_d(G;k)$ counts the $k$-colorings of $G$ in which each vertex has at most $d$ neighbors of its own color. We investigate which structural properties of $G$…

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained…

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…

Assume $ k $ is a positive integer, $ \lambda=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $\lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)} L(v) $ can…

Chromatic-choosablility is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color…

We use a polynomial decomposition result by Stapledon to show that the numerator polynomial of the Ehrhart series of an open polytope is the difference of two symmetric polynomials with nonnegative integer coefficients. We obtain a related…

Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of…

We investigate the \textit{group sum chromatic number} ($\gchi(G)$) of graphs, i.e. the smallest value $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge…

In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic…

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…

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…

If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $\pi(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound…