On a combinatorial problem of Erdos, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.Comment: 15 page

Modified Linear Programming and Class 0 Bounds for Graph Pebbling

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.Comment: 19 pages, 8 figure

Thresholds for zero-sums with small cross numbers in abelian groups

For an additive group $\Gamma$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $\Gamma$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_\Gamma$. The cross number of $S$ is defined to be the sum $\sum_{i=1}^k 1/|g_i|$, where $|g_i|$ denotes the order of $g_i$ in $\Gamma$. Call $S$ good if it contains a zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved that if $\Gamma$ is abelian then every length $|\Gamma|$ sequence of its elements is good, generalizing a 1989 result of Lemke and Kleitman that had proved an earlier conjecture of Erd\H{o}s and Lemke. In 1989 Chung re-proved the Lemke and Kleitman result by applying a theorem of graph pebbling, and in 2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize Geroldinger's result. Here we use probabilistic theorems from graph pebbling to derive a sharp threshold version of Geroldinger's theorem for abelian groups of a certain form. Specifically, we prove that if $p_1, \ldots, p_d$ are (not necessarily distinct) primes and $\Gamma_k$ has the form $\prod_{i=1}^d {\mathbb Z}_{p_i^k}$ then there is a function $\tau=\tau(k)$ (which we specify in Theorem 4) with the following property: if $t-\tau\rightarrow\infty$ as $k\rightarrow\infty$ then the probability that $S$ is good in $\Gamma_k$ tends to 1, while if $\tau-t\rightarrow\infty$ then that probability tends to 0