The odd-even invariant for graphs is the graphic version of the odd-even invariant for oriented matroids. Here, simple properties of this invariant are verified, and for certain graphs, including chordal graphs and complete bipartite graphs, its value is determined. The odd-even chromatic polynomial is introduced, its coefficients are briefly studied, and it is shown that the absolute value of ...

Abstract. Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientatio...

We consider the problem of approximating the independence number and the chromatic number of k-uniform hypergraphs on n vertices. For xed integers k 2, we obtain for both problems that one can achieve in polynomial time approximation ratios of at most O(n=(log 1) n)2). This extends results of Boppana and Halld orsson [5] who showed for the graph case that an approximation ratio of O(n=(logn)) c...

In this paper we develop new algorithmic machinery for solving hard problems on graphs of bounded rank-width and on digraphs of bounded bi-rank-width in polynomial (XP, to be precise) time. These include, particularly, graph colouring and chromatic polynomial problems, the Hamiltonian path and c-min-leaf outbranching, the directed cut, and more generally MSOL-partitioning problems on digraphs. ...

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, wh...

The generalized theta graph s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : :; s k 1. We prove that the roots of the chromatic polynomial ((s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz ? 1j 1 + o(1)] k= logk, uniformly in the path lengths s i. Moreover, we prove that 2;:::;2 ' K 2;k indeed has a chromatic r...

In this section we will discuss the Inclusion-Exclusion principle, with a few applications (including a formula for the chromatic polynomial of a graph), and then consider a wide generalisation of it due to Gian-Carlo Rota, involving the Möbius function of a partially ordered set. The q-binomial theorem gives a simple formula for the Möbius function of the lattice of subspaces of a vector space.

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial ∑n k=1 αk(G)x(x− 1) · · · (x−k+1) of degree n, where αk(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be ∑n k=1 αk(G)x , where G is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge-path and bridge-cycle graphs. Conseque...

In response to a problem of Voloshin, we find recursive formulae for the chromatic polyno-mials of complete r-uniform interval hypergraphs and cohypergraphs. We also give recursive formulae for the chromatic polynomials of complete 3-uniform and 4-uniform interval bihy-pergraphs and comment on the challenges for general r. Our method is to exploit the uniform and complete structure of these hyp...

We study the number P(G) of non-equivalent ways of coloring a given graph G. We show some similarities and differences between this graph invariant and the well known chromatic polynomial. Relations with Stirling numbers of the second kind and with Bell numbers are also given. We then determine the value of this invariant for some classes of graphs. We finally study upper and lower bounds on P(...