For a graphG, let P(G) be its chromatic polynomial. Two graphsG andH are chromatically equivalent if P(G)=P(H). A graph G is chromatically unique if P(H)= P(G) implies that H G. In this paper, we classify the chromatic classes of graphs obtained from K2,2,2 ∪ Pm(m 3), (K2,2,2 − e) ∪ Pm(m 5) and (K2,2,2 − 2e) ∪ Pm(m 6) by identifying the end-vertices of the path Pm with any two vertices of K2,2,...

Let P.G; / be the chromatic polynomial of a graph G. A graph G is chromatically unique if for

Let P (G, λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P (G, λ) = P (H, λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 6-partite graphs with 6n vertices according to the number of 7-independent partitions of G. Using these re...

Let G be a graph of order n, maximum degree ∆ and minimum degree δ. Let P (G,λ) be the chromatic polynomial of G. It is known that the multiplicity of zero ‘0’ of P (G,λ) is one if G is connected; and the multiplicity of zero ‘1’ of P (G,λ) is one if G is 2-connected. Is the multiplicity of zero ‘2’ of P (G,λ) at most one if G is 3-connected? In this paper, we first construct an infinite family...

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...

This note presents two results on real zeros of chromatic polynomials. The first result states that if G is a graph containing a q-tree as a spanning subgraph, then the chromatic polynomial P (G,λ) of G has no non-integer zeros in the interval (0, q). Sokal conjectured that for any graph G and any real λ > ∆(G), P (G,λ) > 0. Our second result confirms that it is true if ∆(G) ≥ bn/3c − 1, where ...

Given a minor-closed class of graphs G, what is the infimum of the non-trivial roots of the chromatic polynomial of G ∈ G? When G is the class of all graphs, the answer is known to be 32/27. We answer this question exactly for three minorclosed classes of graphs. Furthermore, we conjecture precisely when the value is larger than 32/27.

Given a graph we show how to construct a family of manifolds whose Euler characteristics are the values of the chromatic polynomial of the graph at various integers. The manifolds are simple generalisations of configuration spaces.

A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.

Suppose (P,-<) is a poset of size n and n: P-~ P is a permutation. We say that n has a drop at x if n(x)~x. Let fie(k) denote the number of n having k drops, 0 <~ k < n, and define the drop polynomial A p(2) by Further, define the incomparability graph I(P) to have vertex set P and edges 0" whenever i and j are incomparable in P, i.e., neither i-<j nor j< i holds. In this note we give a short p...