Chromatic Polynomials and Bialgebras of Graphs

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

Discrete Mathematics, 2020

Let G be a graph, and let χG be its chromatic polynomial. For any nonnegative integers i, j, we give an interpretation for the evaluation χ (i) G (−j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Greene and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.

The chromatic polynomials are studied by several authors and have important applications in di?erent frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the coe?fficients of the chromatic polynomial of a graph. Three applications on restricted Stirling numbers of the second kind are given.

Discrete Mathematics &amp; Theoretical Computer Science, 2003

We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley&#39;s chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

AKCE International Journal of Graphs and Combinatorics, 2019

The connection between the matching polynomial and the chromatic polynomial for triangle-free graphs was revealed in the work of Farrell and Whitehead. We extend this result to all graph by mirroring the corresponding result of Godsil and Gutman for the acyclic polynomial and the characteristic polynomial. We also reintroduce the clique polynomial of Farrell as an evaluation of the U-polynomial of Noble and Welsh, which also happens to contain the matching and chromatic polynomials. c

IJCRM.COM, 2016

This article is a general introduction of chromatic polynomials.In this,Chromatic polynomials are defined,their connection between the theory of chromatic polynomials and coloring of graphs.Also it explains the chromatic polynomials of total graphs .It gives the basic concepts of chromatic polynomials in graph theory.

Linear Algebra and its Applications, 2002

The chromatic polynomials considered in this paper are associated with graphs constructed in the following way. Take n copies of a complete graph K b and, for i = 1, 2, . . . , n, join each vertex in the ith copy to the same vertex in the (i + 1)th copy, taking n + 1 = 1 by convention. Previous calculations for b = 2 and b = 3 suggest that the chromatic polynomial contains terms that occur in 'levels'. In the present paper the levels are explained by using a version of the sieve principle, and it is shown that the terms at level correspond to the irreducible representations of the symmetric group Sym . In the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance-regular graphs. For the nonlinear representations the calculations are more complicated. An illustration is given in Section 10, where the complete chromatic polynomial for the case b = 4 is obtained. MSC 2000: 05C15, 05C50.

Theory and applications of graphs, 2022

For m ≥ 3 and n ≥ 1, the m-cycle book graph B(m, n) consists of n copies of the cycle C m with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed B(m, n) is n + 1, and (b) the chromatic number of a signed B(m, n) is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.

2015

The chromatic polynomial, PΓ(x) of a graph Γ, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph Γ. We can then find the orbital chromatic polynomial OPΓ,G(x) of a graph Γ and a group G of automorphisms of Γ, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph Γwhen acted upon by the group G. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture concerning the gap between the real roots of the chromatic polynomial and the real roots of the orbital chromatic polynomial.

Discrete Mathematics &amp; Theoretical Computer Science

Combinatorics A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.