David Galvin

Two problems on independent sets in graphs

Let i t ( G ) denote the number of independent sets of size t in a graph G . Levit and Mandrescu have conjectured that for all bipartite G the sequence ( i t ( G ) ) t ? 0 (the independent set sequence of G ) is unimodal. We provide evidence for this conjecture by showing that this is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph G ( n , n , p ) , and show that for any fixed p ? ( 0 , 1 ] its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for p = ? ? ( n - 1 / 2 ) .We also consider the problem of estimating i ( G ) = ? t ? 0 i t ( G ) for G in various families. We give a sharp upper bound on the number of independent sets in an n -vertex graph with minimum degree ? , for all fixed ? and sufficiently large n . Specifically, we show that the maximum is achieved uniquely by K ? , n - ? , the complete bipartite graph with ? vertices in one partition class and n - ? in the other.We also present a weighted generalization: for all fixed x 0 and ? 0 , as long as n = n ( x , ? ) is large enough, if G is a graph on n vertices with minimum degree ? then ? t ? 0 i t ( G ) x t ? ? t ? 0 i t ( K ? , n - ? ) x t with equality if and only if G = K ? , n - ? . Highlights? The stable set sequence of the random equibipartite graph is almost surely unimodal. ? Apart from perhaps a short initial segment, it is almost surely log-concave. ? The n vertex graph with minimum degree d with the most stable sets is K ( d , n - d ) .

On homomorphisms from the Hamming cube to Z

AbstractWriteF for the set of homomorphisms from {0, 1}d toZ which send0 to 0 (think of members ofF as labellings of {0, 1}d in which adjacent strings get labels differing by exactly 1), andF1 for those which take on exactlyi values. We give asymptotic formulae for |F| and |F|.In particular, we show that the probability that a uniformly chosen memberf ofF takes more than five values tends to 0 asd→∞. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constantb such thatf a.s. takes at mostb values. This in turn verified a conjecture of I. Benjaminiet al., that for eacht>0,f a.s. takes at mosttd values.Determining |F| is equivalent both to counting the number of rank functions on the Boolean lattice 2[d] (functionsf: 2[d]→N satisfyingf(

Phase coexistence and torpid mixing in the 3-coloring model on Z^d

f\not 0 = 0

Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2

) andf(A)≤f(A∪x)≤f(A)+1 for allA∈2[d] andx∈[d]) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from {0, 1}d toK3, the complete graph on 3 vertices).Our proof uses the main lemma from Kahn’s proof of constant range, together with some combinatorial approximation techniques introduced by A. Sapozhenko.

The Number of Independent Sets in a Graph with Small Maximum Degree

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H-colorings admitted by an n-vertex, d-regular graph, for each H. Specifically, writing for the number of H-colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n-vertex, d-regular, loopless graph G, we have where is the complete bipartite graph with d vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of H. This allows us to characterize those H for which is eventually (for all sufficiently large d) larger than and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d-regular G, we have where as . More precise results are obtained in some special cases.

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

We show that for all sufficiently large

H-colouring bipartite graphs

d

Extremal H-colorings of trees and 2-connected graphs

, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on