For a given graph G, let P (G,λ) be the chromatic polynomial of G, where λ is considered to be a real number. In this paper, we study the bounds for P (G,λ)/P (G,λ − 1) and P (G,λ)/P (G − x, λ), where x is a vertex in G, λ ≥ n and n is the number of vertices of G.

Let H = (X,E) be a simple hypergraph and let f (H, ) denote its chromatic polynomial. Two hypergraphs H1 and H2 are chromatic equivalent if f (H1, ) = f (H2, ). The equivalence class of H is denoted by 〈H 〉. Let K and H be two classes of hypergraphs.H is said to be chromatically characterized in K if for every H ∈ H ∩K we have 〈H 〉 ∩K=H ∩K. In this paper we prove that uniform hypertrees and uni...

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. HelmeGuizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only non-trivial cohomology groups H(Γ), H(Γ) come in isomorphic pairs: H(Γ) = H(Γ) for i > 0 if Γ is non-bipartite, and for i > 0 if Γ is bipartite. As a corollary, the ranks of the cohomolog...

For a graph G, we denote by P (G, λ) the chromatic polynomial of G and by h(G, x) the adjoint polynomial of G. A graph G is said to be chromatically unique if for any graph H, P (H, λ) = P (G, λ) implies H ∼= G. In this paper, we investigate some algebraic properties of the adjoint polynomials of some graphs. Using these properties, we obtain necessary and sufficient conditions for Kn − E(∪a,bT...

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homolog...

Let P(G;x,y) be the number of vertex colorings φ : V →{1, ...,x} of an undirected graph G = (V,E) such that for all edges {u,v} ∈ E the relations φ(u)≤ y and φ(v)≤ y imply φ(u) 6= φ(v). We show that P(G;x,y) is a polynomial in x and y which is closely related to Stanley’s chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, an...

It is known that the chromatic polynomial of any chordal graph has only integer roots. However, there also exist non-chordal graphs whose chromatic polynomials have only integer roots. Dmitriev asked in 1980 if for any integer p¿ 4, there exists a graph with chordless cycles of length p whose chromatic polynomial has only integer roots. This question has been given positive answers by Dong and ...

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar...