We prove that if an edge of a cycle on n vertices is extended by adding k vertices, then the the chromatic polynomial of such generalized cycle is: P (Hk, λ) = (λ− 1) k ∑ i=0 λ + (−1)n(λ− 1).

A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are ana...

A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are ana...

For a graph G, let P(G; ) be its chromatic polynomial and let [G] be the set of graphs having P(G; ) as their chromatic polynomial. We call [G] the chromatic equivalence class of G. If [G]={G}, then G is said to be chromatically unique. In this paper, we 4rst determine [G] for each graph G whose complement 5 G is of the form aK1∪bK3∪⋃16i6s Pli , where a; b are any nonnegative integers and li is...

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w ∈ Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of thi...

In this short note we observe that recent results of Abért and Hubai and of Csikvári and Frenkel about Benjamini–Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.

Abstract In this paper we obtain the explicit formulas for chromatic polynomials of cacti. From the results relating to cacti we deduce the analogous formulas for the chromatic polynomials of n–gon–trees. Besides, we characterize unicyclic graphs by their chromatic polynomials. We also show that the so–called clique–forest–like graphs are chromatically equivalent.

In this paper we introduce multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, a...

In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial PG(q) of a finite graph G of maximal degree D lie in the disc |q| < KD, where K is a constant that is strictly smaller than 8.