Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

TL;DR

This paper determines optimal addressings for Johnson graphs and explores addressing problems for other graphs, providing bounds, computational data, and settling a related open problem in graph addressing.

Contribution

It introduces an optimal addressing scheme for Johnson graphs and analyzes addressing lengths for various graphs, including complete multipartite graphs and odd cycles.

Findings

Optimal addressing length for Johnson graphs $J(n,k)$ is $k(n-k)$.

Addressing is optimal for $k=1$ and for $k=2$, $n=4,5,6$, but not for $n=6$, $k=3$.

Most graphs on $n$ vertices have addressings of length at most $n-(2-o(1)) ext{log}_2 n$.

Abstract

Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))\log_2 n$.