How to burn a Latin square

TL;DR

This paper studies the lazy burning process on hypergraphs associated with Latin squares, establishing bounds on the lazy burning number and exploring its properties through various Latin square structures.

Contribution

It introduces bounds for the lazy burning number of Latin square hypergraphs and analyzes their tightness, also examining specific cases like cyclic Latin squares and group-derived hypergraphs.

Findings

Bounds for the lazy burning number are tight for certain Latin squares.

Lazy burning number relates to the structure of Latin squares and their subsquares.

Explicit values are determined for hypergraphs from finitely generated groups.

Abstract

We investigate the lazy burning process for Latin squares by studying their associated hypergraphs. In lazy burning, a set of vertices in a hypergraph is initially burned, and that burning spreads to neighboring vertices over time via a specified propagation rule. The lazy burning number is the minimum number of initially burned vertices that eventually burns all vertices. The hypergraphs associated with Latin squares include the $n$-uniform hypergraph, whose vertices and hyperedges correspond to the entries and lines (that is, sets of rows, columns, or symbols) of the Latin square, respectively, and the $3$-uniform hypergraph, which has vertices corresponding to the lines of the Latin square and hyperedges induced by its entries. Using sequences of vertices that together form a vertex cover, we show that for a Latin square of order $n$, the lazy burning number of its $n$-uniform hypergraph is bounded below by $n^2-3n+3$ and above by $n^2-3n+2 + \lfloor \log_2 n \rfloor.$ These bounds are shown to be tight using cyclic Latin squares and powers of intercalates. For the $3$-uniform hypergraph case, we show that the lazy burning number of Latin squares is one plus its shortest connected chain of subsquares. We determine the lazy burning number of Latin square hypergraphs derived from finitely generated groups. We finish with open problems.