Nancy Eaton

Ramsey numbers for sparse graphs

Abstract We consider a class of graphs on n vertices, called ( d , f )-arrangeable graphs. This class of graphs contains all graphs of bounded degree d , and all df -arrangeable graphs, a class introduced by Chen and Schelp in 1993. In 1992, a variation of the Regularity Lemma of Szemeredi was introduced by Eaton and Rodl. As an application of this lemma, we give a linear upper bound, c ( d , f ) n , for the Ramsey number of graphs in this class, where log 2 log 2 c ( d , f ) = 24 df 5 . This improves the earlier result, given in 1983 by Chvatal et al. of a linear bound on the Ramsey number of graphs with bounded degree d , where the constant term was more that a tower of d 2s, and later extended by Chen and Schelp to include d -arrangeable graphs.

On pebbling threshold functions for graph sequences

Given a connected graph G, and a distribution of t pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number t, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs g = (G1, G2,...,Gn,...), where Gn has n vertices, is any function t0(n) such that almost all distributions of t pebbles are solvabe when t » t0, and such that almost none are solvable when t « t0. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.

Graphs of small dimensions

AbstractLetG=(V, E) be a graph withn vertices. The direct product dimension pdim (G) (c.f. [10], [12]) is the minimum numbert such thatG can be embedded into a product oft copies of complete graphsKn.In [10], Lovász, Nešetřil and Pultr determined the direct product dimension of matchings and paths and gave sharp bounds for the product dimension of cycles, all logarithmic in the number of vertices.Here we prove that pdim (G)≤cdlogn for any graph with maximum degreed andn vertices and show that up to a factor of

A Canonical Ramsey Theorem

1/\left( {\log d + \log \log \frac{n}{{2d}}} \right)

On the Erdős-Sós conjecture for graphs having no path with k+4 vertices

this bound is the best possible.We also study set representations of graphs. LetG=(V,E) be a graph andp≥1 an integer. A familyF={Ax,x∈V} of (not necessarily distinct) sets is called ap-intersection representation ofG if |Ax↓Ay|≥p⇔{x,y}∈E for every pairx, y of distinct vertices ofG. Let θG(G) be the minimum size of |UF| taken over all intersection representations ofG. We also study the parameter θ(G)=

On p -intersection representations

\mathop {\min }\limits_p