Distance labelings: a generalization of Langford sequences

TL;DR

This paper introduces two new generalizations of Langford sequences, expanding the concept of distance labelings on paths, which could have implications for combinatorial design and graph labeling problems.

Contribution

The paper presents novel generalizations of Langford sequences related to distances, broadening the scope of sequence labelings in combinatorics.

Findings

Two new types of distance labelings introduced

Connections established between labelings and graph structures

Potential applications in combinatorial design and graph theory

Abstract

A Langford sequence of order $m$ and defect $d$ can be identified with a labeling of the vertices of a path of order $2m$ in which each labeled from $d$ up to $d+m-1$ appears twice and in which the vertices that have been label with $k$ are at distance $k$. In this paper, we introduce two generalizations of this labeling that are related to distances.