Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G∼H , if P(G) = P(H). A graph G is chromatically unique if for any graph H , G∼H implies that G is isomorphic with H . In this paper, we give the necessary and su:cient conditions for a family of generalized polygon trees to be chromatically unique. c © 2001 Elsevier Science B.V. All ...

For any positive integer n, let Gn denote the set of simple graphs of order n. For any graph G in Gn, let P(G; ) denote its chromatic polynomial. In this paper, we -rst show that if G ∈Gn and (G)6 n− 3, then P(G; ) is zero-free in the interval (n − 4 + =6 − 2= ;+∞), where = (108 + 12√93)1=3 and =6 − 2= (=0:682327804 : : :) is the only real root of x + x − 1; we proceed to prove that whenever n ...

The chromatic polynomial P (G,λ) gives the number of λ-colourings of a graph. If P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ), then the graph G is said to have a chromatic factorisation with chromatic factors H1 and H2. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic fac...

The chromatic polynomial gives the number of proper λ-colourings of a graph G. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ) for some graphs H1 and H2 and clique Kr. It is known that the c...

In this note we consider s-chromatic polynomials for finite simplicial complexes. When s = 1, the 1-chromatic polynomial is just the usual graph chromatic polynomial of the 1-skeleton. In general, the s-chromatic polynomial depends on the s-skeleton and its value at r is the number of (r, s)-colorings of the simplicial complex.

Whitney’s Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets. We give a definition of cycles in hypergraphs that preserves the statement of the theorem there.

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, if P(G) = P(H). A set of graphs S is called a chromatic equivalence class if for any graph H that is chromatically equivalent with a graph G in S, then H ∈S. Peng et al. (Discrete Math. 172 (1997) 103–114), studied the chromatic equivalence classes of certain generalized polygon trees. In thi...