A. J. Radcliffe

Extremal graphs for homomorphisms

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P**math-image**, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright

Reconstructing Subsets of Zn

In this paper we consider the problem of reconstructing a subsetA?Zn, up to translation, from the collection of its subsets of sizek, given up to translation (itsk-deck). Results of Alon, Caro, Krasikov, and Roditty (1989,J. Combin. Theory Ser. B47, 153?161) show that this is possible providedk>log2n. Mnukhin (1992,Acta. Appl. Math.29, 83?117) showed that every subset of Znof sizekis reconstructible from its (k?1)-deck, providedk?4. We show that whennis prime every subset of Znis reconstructible from its 3-deck; that for arbitrarynalmost all subsets of Znn are reconstructible from their 3-decks; and that for anynevery subset of Znis reconstructible from its 9?(n)-deck, where?(n) is the number of distinct prime factors ofn. We also comment on analogous questions for arbitrary groups, in particular the cube Zn2. Our approach is to generalize the problem to that of reconstructing arbitrary rational functions on Zn. We prove?by analysing the interaction between the ideal structure of the group ring QZnand the operation of pointwise multiplication of functions?that with a suitable definition of deck every rational-valued function on Znis reconstructible from its 9?(n)-deck.

Minimizing the number of independent sets in triangle-free regular graphs

Abstract Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a d -regular graph is maximized by disjoint copies of K d , d . Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.

On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph G is the maximum density of an independent set in G . Lih etźal. (1999) showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs.We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three and determine asymptotic values for others.

Extremal Cases of the Ahlswede-Cai Inequality

Consider two set systems A and B in the powerset P(n) with the property that for eachA?A there exists a uniqueB?B such thatA?B. Ahlswede and Cai proved an inequality about such systems which is a generalization of the LYM and Bollobas inequalities. In this paper we characterize the structure of the extremal cases.

Extremal Graphs for Homomorphisms II: EXTREMAL GRAPHS FOR HOMOMORPHISMS II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let hom(G,H) denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize hom(G,H). We prove that if H is loop-threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges which maximizes hom(G,H). Similarly, we show that loop-quasi-threshold image graphs have quasi-threshold extremal graphs. In the case H = P o 3 , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for hom(G,P o 3 ). Also in this paper, using similar techniques, we determine a set of extremal graphs for “the fox”, a graph formed by deleting the loop on one of the end-vertices of P o 3 . The fox is the unique connected loop-threshold image graph on at most three vertices for which the extremal problem was not previously solved.

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs

Graphs with the fewest matchings

G

The Multistep Friendship Paradox

with

Counting dominating sets and related structures in graphs

n