Leaper Embeddings

TL;DR

This work analyzes how large a grid graph $\square_m$ can be embedded into the leaper graphs $\mathcal{L}_n$ formed by a skew leaper with step $(p,q)$, assuming $\gcd(p,q)=1$. It distinguishes free leapers (opposite-parity $p$ and $q$) where $m = n + O_L(1)$ and half-free leapers (both odd) where $m = n/2 + O_L(1)$, with the latter case requiring novel combinatorial-geometry tools based on forcing chords. The main approach blends constructive embedding schemes for both cases with a new forced-chord framework (including both single-vector and disjunctive variants) and a lattice-theoretic upper bound in the half-free setting. The results illuminate how grid-like structures can be embedded in chess-piece move graphs and connect discrete geometry with chord-forcing phenomena, offering techniques potentially applicable to broader embedding and combinatorial-geometry problems.

Abstract

A leaper is a chess piece which generalises the knight. Given $n$ and a $(p, q)$-leaper $L$, we study the greatest $m$ such that the $m \times m$ grid graph can be embedded into the $n \times n$ leaper graph of $L$. We can assume that $p$ and $q$ are relatively prime. We show that $m \approx n$ when $p$ and $q$ are of opposite parities and $m \approx n/2$ otherwise. The latter case is substantially more difficult. The proof involves certain combinatorial-geometric results on the chords of connected figures which might be of independent interest.

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