Related papers: Bivariate Chromatic Polynomials of Mixed Graphs

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed…

This paper discusses ways to categorify chromatic, dichromatic and Penrose polynomials, including categorifications of integer evaluations of chromatic polynomials. We show that with an appropriate choice of variables the coefficients of…

This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a basis given by chromatic symmetric…

Suppose $G$ is a simple graph with $n$ vertices, $m$ edges, and rank $r$. Let $\chi_G(t)=a_0t^n-a_1t^{n-1}+\cdots +(-1)^ra_rt^{n-r}$ be the chromatic polynomial of $G$. For $q,k\in \Bbb{Z}$ and $0\le k\le q+r+1$, we obtain a sharp two-side…

By a graph we mean a finite undirected graph having multiple edges but no loops. Given a graph property $\mathcal{P}$, a $\mathcal{P}$-coloring of a graph $G$ with color set $C$ is a mapping $\f:V(G)\to C$ such that for each color $c\in C$…

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph…

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic…

A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is…

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced…

We prove that for every path H, and every integer d, there is a polynomial f such that every graph G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains as a subgraph the complete d-partite graph…

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…

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain…

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees,…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if…

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. The chromatic polynomial of a graph is an…

Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with a same color, if there is a shortest $x,y$-path whose internal vertices have different colors than $x,y$. The smallest…

Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we…

We prove that for every $t\in \mathbb{N}$ there is a constant $\gamma_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{\gamma_t \log^{4t+3} \omega}$. In other words, we…

In the present note we show, via the connection between chromatic polynomial and Potts model, that the Whitney Broken circuit theorem is in fact a special case of a more general identity relating the chromatic polynomial of a graph G=(V,E)…