Maximal 2-dimensional binary words of bounded degree

TL;DR

This work addresses the problem of maximizing the number of filled cells in an $h\times w$ 2D word over a binary alphabet under a degree bound $d$ on filled cells, formalized as $\mathrm{max}_{\le d}(h,w)$. It introduces the excess function $e(W)=|W|_{\blacksquare}-\tfrac{2}{3}hw$ and the notion of $2$-full words to study the case $d=2$, proving the exact bound $\mathrm{max}_{\le d}(h,w)=m_{\le d}(h,w)$ with explicit piecewise formulas for $m_{\le d}(h,w)$ in each $d$-case, including a domination-number interpretation for $d=3$. The analysis proceeds by handling $d\in\{0,1,4\}$ directly, relating $d=3$ to grid-dominating sets, and then performing a detailed constructive and extremal study for $d=2$, with base cases up to width $6$ and a general-width construction via $Q$-shapes and concatenations. Consequences include tight bounds for maximal snakes in $h\times w$ rectangles and insights into potential extensions to polycubes and higher-dimensional languages; several open questions remain regarding uniqueness of extremal words and exact snake lengths in broader regimes. The results advance pattern-avoidance and grid-graph domination perspectives within multidimensional word combinatorics and provide exact formulas useful for related polyomino and sequence-avoidance analyses.

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.

Figures (4)

• Figure 1: (a) A $7 \times 4$ word $W$ on the alphabet $\{\square,\blacksquare\}$. (b) The $\blacksquare$-degree word of $W$. (c) The geometric representation of $W$. (d) The grid graph $G_{7,4}$. (e) The subgraph (in green) of $G_{7,4}$ (in pale gray) induced by $W$. Row and column indices of $W$ appear in blue.
• Figure 2: $7 \times 4$$d$-full words, for (a) $d = 0$, (b) $d = 1$, (c) $d = 2$, (d) $d = 3$ and (e) $d = 4$.
• Figure 3: The word $W = \left(\substack{\blacksquare\square\\\square\blacksquare}\right)^{h/2 \times w/2}$ superimposed with an arbitrary dominoes/monomino tiling, when the dimensions are (a) $6 \times 6$ (b) $5 \times 5$.
• Figure 4: Some 2-full words

Theorems & Definitions (28)

• theorem thmcountertheorem
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof : sketched
• lemma thmcounterlemma
• proof
• definition thmcounterdefinition: Excess of a word
• proposition thmcounterproposition
• theorem thmcountertheorem