Comments on “Extremal Cayley Digraphs of Finite Abelian Groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]

We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.Peer ReviewedPostprint (published version

Answers to two questions posed by Farhi concerning additive bases

Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.Comment: 7 pages, no figures. This is v3 : I found a gap in the proof of Lemma 3.2 of v2. This has now been corrected and the same result is Lemma 3.3 in this versio

Extremal bases for finite cyclic groups

AbstractLet m and h be positive integers. A set A of integers is called a basis of orderh for Z(m) if every integer n is congruent to a sum of h elements in A modulo m. Let m(h, A) denote the greatest positive integer m such that A is a basis of order h for Z(m). For any k ≥ 1, define m(h, k) = max∥A∥ = k + 1 m(h, A). This generalizes a function of Graham and Sloane. In this paper, it is proved that, for fixed k ≥ 4 as h → ∞, m(h, k) ≥ αk (256125)⌞k4⌟ (hk)k + O(hk − 1), where αk = 1 if k ≡ 0 or 1 (mod 4), 43 if k ≡ 2 (mod 4), and 2716 if k ≡ 3 (mod 4). A lower bound for m(h, k) is also obtained for fixed h. Using these results, new lower bounds are proved for the order of subsets of asymptotic bases

Cayley digraphs of finite abelian groups and monomial ideals

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph’s diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals

Additive Combinatorics: A Menu of Research Problems

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.Comment: This August 2017 version incorporates feedback and updates from several colleague