We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider squareand triangular-lattice strips with fixed width L, arbitrary length N , and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L = 5, 6, 7 for the...

We consider the problem of approximating the b-chromatic number of a graph. We show that there is no constant ε > 0 for which this problem can be approximated within a factor of 120/113− ε in polynomial time, unless P = NP. This is the first hardness result for approximating the b-chromatic number.

We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m ≤ 12P, 9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such...

The chromatic polynomial PG(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (−∞, 0), (0, 1) and (1, 32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multi...

In this paper, we generalize the classical Bonferroni inequalities and their improvements by Galambos to sums of type ∑ I⊆U (−1)|I|f(I) where U is a finite set and f : 2 → R. The result is applied to the Tutte polynomial of a matroid and the chromatic polynomial of a graph.

Recall that a polynomial P (x) ∈ Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients −1, 1 is called a Littlewood polynomial. Is there an algebraic number α which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is ...

We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m ≤ 12P, 9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such...

For a hypergraphic polytope there is weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines f-polynomial. This invariant of hypergraphs extends the Stanley chromatic symmetric simple graphs. We consider certain combinatorial Hopf algebra show that universal morphism to functions coincides with this enumerator function. calculate f-polynomial u...