Slitherlink on Triangular Grids

TL;DR

This paper characterizes totally even subsets and their properties in the triangular grid, revealing divisibility conditions on the symmetric difference of cycles with the same signature, advancing understanding of combinatorial structures in grid graphs.

Contribution

It provides a complete characterization of totally even subsets in the triangular grid and establishes divisibility properties of cycle differences based on signatures.

Findings

Totally even subsets are fully characterized in the triangular grid.

The size of symmetric differences of cycles with the same signature is divisible by 12.

The divisibility bound of 12 is proven to be optimal.

Abstract

Let $G$ be a plane graph and let $C$ be a cycle in $G$. For each finite face of $G$, count the number of edges of $C$ the face contains. We call this the Slitherlink signature of $C$. The symmetric difference $A$ of two cycles with the same signature is totally even, meaning every vertex is incident to an even number of edges in $A$ and every face contains an even number of edges in $A$. In this paper, we completely characterize totally even subsets in the triangular grid, and count the number of edges in any totally even subset of the triangular grid. We also show that the size of the symmetric difference of two cycles with the same signature in the triangular grid is divisible by $12$; this is best possible since 12 is the greatest common divisor of all the sizes of the symmetric difference between two cycles with the same signature in a triangular grid.