Woodall, D.R., An inequality for chromatic polynomials, Discrete Mathematics 101 (1992) 327-331. It is proved that if P(G, t) is the chromatic polynomial of a simple graph G with II vertices, m edges, c components and b blocks, and if t S 1, then IP(G, t)/ 2 1t’(t l)hl(l + ys + ys2+ . + yF’ +spl), where y = m n + c, p = n c b and s = 1 t. Equality holds for several classes of graphs with few ci...

Let = (m 1 ; : : : ; m k) denote a graph consisting of two vertices u; v and k internally-disjoint uv-paths of lengths m 1 , : : : ,m k. We exhibit an implicitly-deened function R(k) such that every complex zero z of the chromatic polynomial of satisses jz ? 1j < R(k), and show that R(k) is sublinear as k ! 1.

Let the vertices of a graph such that every two adjacent vertices have different color is a very common problem in the graph theory. This is known as proper coloring of graphs. The possible number of different proper colorings on a graph with a given number of colors can be represented by a function called the chromatic polynomial. Two graphs G and H are said to be chromatically equivalent, if ...

In a paper from 1912 aimed at proving the four-color theorem, G. D. Birkhoff [Birkhoff 12] introduced a function P (G, x), defined for all positive integers x to be the number of proper x-colorings of the graph G. As it turns out, P (G, x) is a polynomial in x and so is defined for all real and complex values of x as well. Of course, P (G, x) is the by now well-known chromatic polynomial, and a...

Although algebraic or so-called “implicit polynomial” curves have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper develops a dynamic formulation for implicit polynomial curves based on level set formalism. In particular, it is shown that utilization of an implicit poly...

The chromatic polynomial PΓ(x) of a graph Γ is a polynomial whose value at the positive integer k is the number of proper kcolourings of Γ. If G is a group of automorphisms of Γ, then there is a polynomial OPΓ,G(x), whose value at the positive integer k is the number of orbits of G on proper k-colourings of Γ. It is known there are no real negative chromatic roots, but they are dense in [ 27 ,∞...

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.

In this paper, all graphs considered are simple graphs. We always suppose that G is a graph. Let V(G), E(G), v(G), and e(G) be the vertex set, edge set, order and number of edges of G. For a positive integer *, a *-colouring of G is a mapping f : V(G) [1, ..., *] such that f (x){ f ( y) whenever x and y are adjacent in G. Let P(G, *) denote the number of *-colourings in G. It is well known that...

A family (St) of sets is p-bounded Diophantine if St has a representing p-bounded polynomial RS;t, s.t. x 2 St () (9y)[RS(x; y) = 0℄. We say that (St) is unbounded Diophantine if additionally, RS;t is a fixed t-independent polynomial. We show that p-bounded (resp., unbounded) Diophantine set has a polynomial-size (resp., constant-size) statistical zero-knowledge proof system that a committed tu...