arXiv subjects

Haran Mouli

Publications and source records attributed to Haran Mouli.

4 recordsLinked to original sources

arXiv2026

Grid designs

We define a grid graph $G$ as a Cartesian product of path-graphs $P_n$ or cycle-graphs $C_n$ as shown in Figure 1, and we ask, when can the edge set of a complete graph be expressed as a disjoint union of graphs isomorphic to $G$? That is, we are asking for which grid graphs a $G$-design exists, where a $G$-design is defined as a decomposition of a complete graph into edge-disjoint subgraphs isomorphic to $G$. We show that when $n$ is an odd prime or the square of an odd prime, the toroidal grid-graph $G = C_n \square C_n$ admits a $G$-design. In the less symmetrical case of products of path-graphs, we prove that $G = P_3 \square P_3$ does not admit a $G$-design but that $G = P_4 \square P_4$ does. This last result is the special case that motivated the present paper: a $P_4 \square P_4$-design corresponds to a way of successively scrambling a Connections puzzle so that each pair of words occurs adjacently exactly once. Our constructions use the arithmetic of finite fields.

math.CO↗

arXiv2025

The Index Problem for Subgroup Intersections

In previous papers, Drungilas et al. study the problem of which triples of positive integers $(a, b, c)$ can be realized as $([E: \mathbb{Q}], [F: \mathbb{Q}], [EF: \mathbb{Q}])$, where $E$ and $F$ are number fields, using techniques from field theory. We shall study this problem rephrased in the language of groups using the Galois correspondence to simplify and generalize their results.

math.GR↗

arXiv2025

Classification of Finite Groups With Equal Left and Right Quotient Sets

In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A \subseteq G$, we have $|AA^{-1}| = |A^{-1}A|$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].

math.GR↗

arXiv2022

On the Existence of Balanced Generalized de Bruijn Sequences

A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine necessary and sufficient conditions on $n,l$, and $k$ for the existence of such a sequence.

math.CO↗