Induction on Descent in Leaper Graphs

TL;DR

This paper introduces a novel inductive method called induction on descent to analyze leaper graphs, constructing an infinite tree structure to relate various leapers and prove properties about their graphs.

Contribution

It develops a new proof technique that systematically relates leaper graphs through transformations, enabling comprehensive analysis of skew free leapers.

Findings

Established theorems about all skew free leapers

Developed a recursive framework for leaper graph analysis

Resolved several open questions in leaper graph theory

Abstract

We construct an infinite ternary tree $\mathfrak{L}$ whose root is the knight and whose vertices are all skew free leapers. We define the descent of a skew free leaper to be its "address" within $\mathfrak{L}$. We introduce three transformations which relate the leaper graphs of a skew free leaper to the leaper graphs of its three children in $\mathfrak{L}$. By starting with the knight and then applying these transformations so as to advance throughout $\mathfrak{L}$, we can establish theorems about all skew free leapers. We call this proof technique induction on descent and with its help we resolve a number of questions about leaper graphs.