The packing number of the double vertex graph of the path graph

TL;DR

This paper determines the packing number of the double vertex graph of a path, solving an open problem and confirming a conjecture related to binary codes and graph theory.

Contribution

It provides the first exact calculation of the packing number for the double vertex graph of a path, linking it to a known graph and resolving a longstanding open problem.

Findings

Calculated the packing number of the double vertex graph of a path.

Confirmed Rob Pratt's conjecture on the generating function of sequence A085680.

Connected the problem to binary codes correcting adjacent transpositions.

Abstract

Neil Sloane showed that the problem of determine the maximum size of a binary code of constant weight 2 that can correct a single adjacent transposition is equivalent to finding the packing number of a certain graph. In this paper we solve this open problem by finding the packing number of the double vertex graph (2-token graph) of a path graph. This double vertex graph is isomorphic to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the ordinary generating function of sequence A085680.