Let ) , ( λ χ G denotes the number of proper vertex colourings of with at most G λ colours. G. Birkhoff , observed in 1912 that ) , ( λ χ G is, for a fixed graph , a polynomial in G λ , which is now called the chromatic polynomial of . More precisely, let G be a simple graph and G N ∈ λ . A mapping } , {1,2, ) ( : λ K → G V f is called a λ -colouring of if whenever the vertices and are adjacent...

Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion–exclusion and the zeta transform. In ...

Let T be an unrooted tree. The chromatic symmetric function XT , introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of T . The subtree polynomial ST , first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of T by their numbers of edges and leaves. We prove that ST = 〈Φ,XT 〉, where 〈·, ·〉 ...

We consider a graph polynomial ξ(G;x, y, z) introduced by Ilia Averbouch, Benny Godlin, and Johann A. Makowsky (2008). This graph polynomial simultaneously generalizes the Tutte polynomial as well as a bivariate chromatic polynomial defined by Klaus Dohmen, André Pönitz, and Peter Tittmann (2003). We derive an identity which relates the graph polynomial ξ of a thickened graph (i.e. a graph with...

The chromatic polynomial P (G, λ) gives the number of proper colourings of a graph G in at most λ colours. If P (G, λ) = P (H1, λ)P (H2, λ) /P (Kr, λ), then G is said to have a chromatic factorisation of order r with chromatic factors H1 and H2. It is clear that, if 0 ≤ r ≤ 2, any H1 6∼= Kr with chromatic number χ(H1) ≥ r is the chromatic factor of some chromatic factorisation of order r. We sh...

In this paper, we give an algorithm that computes the chromatic number of a graph in O(5.283n) time and polynomial memory.

Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G. The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisn...

I define models of quantum loops and nets that have ground states with topological order. These make possible excited states comprised of deconfined anyons with non-abelian braiding. With the appropriate inner product, these quantum loop models are equivalent to net models whose topological weight involves the chromatic polynomial. A simple Hamiltonian preserving the topological order is found ...

We determine the isogeny classes of abelian surfaces over Fq whose group of Fq-rational points has order divisible by q . We also solve the same problem for Jacobians of genus-2 curves. In a recent paper [4], Ravnshøj proved: if C is a genus-2 curve over a prime field Fp, and if one assumes that the endomorphism ring of the Jacobian J of C is the ring of integers in a primitive quartic CM-field...