From the Ising and Potts models to the general graph homomorphism polynomial

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte-polynomal.Comment: V2. Extended versio

Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(\Delta^{3/2})$ colors suffice, and acyclic colorings, for which $O(\Delta^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree $\Delta$ has a proper coloring with $O(\Delta^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(\Delta^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.Comment: 6 page

Linear-programming design and analysis of fast algorithms for Max 2-CSP

The class Max (r,2)-CSP, or simply Max 2-CSP, consists of constraint satisfaction problems with at most two r-valued variables per clause. For instances with n variables and m binary clauses, we present an O(nr5+19m/100)-time algorithm which is the fastest polynomial-space algorithm for many problems in the class, including Max Cut. The method also proves a treewidth bound , which gives a faster Max 2-CSP algorithm that uses exponential space: running in time O(2(13/75+o(1))m), this is fastest for most problems in Max 2-CSP. Parametrizing in terms of n rather than m, for graphs of average degree d we show a simple algorithm running time , the fastest polynomial-space algorithm known. In combination with “Polynomial CSPs” introduced in a companion paper, these algorithms also allow (with an additional polynomial factor overhead in space and time) counting and sampling, and the solution of problems like Max Bisection that escape the usual CSP framework. Linear programming is key to the design as well as the analysis of the algorithms