Maximal 2-dimensional binary words of bounded degree

TL;DR

This paper derives an exact formula for the maximum number of 1s in a 2D binary word with bounded degree, and applies this to bound the size of snake polyominoes within rectangles.

Contribution

It provides a precise formula for the maximum number of 1s in 2D binary words with degree constraints, linking combinatorics and geometric properties.

Findings

Exact maximum count of 1s for given dimensions and degree bound

Upper bounds on snake polyominoes in rectangles

Connections between binary words and geometric structures

Abstract

Let d be an integer between 0 and 4, and W be a 2-dimensional word of dimensions h x w on the binary alphabet {0, 1}, where h, w in Z > 0. Assume that each occurrence of the letter 1 in W is adjacent to at most d letters 1. We provide an exact formula for the maximum number of letters 1 that can occur in W for fixed (h, w). As a byproduct, we deduce an upper bound on the length of maximum snake polyominoes contained in a h x w rectangle.