On Disjoint Golomb Rulers

Abstract

A set $\{a_i\:|\: 1\leq i \leq k\}$ of non-negative integers is a Golomb ruler if differences $a_i-a_j$, for any $i \neq j$, are all distinct. A set of $I$ disjoint Golomb rulers (DGR) each being a $J$-subset of $\{1,2,\cdots, n\}$ is called an $(I,J,n)-DGR$. Let $H(I, J)$ be the least positive $n$ such that there is an $(I,J,n)-DGR$. In this paper, we propose a series of conjectures on the constructions and structures of DGR. The main conjecture states that if $A$ is any set of positive integers such that $|A| = H(I, J)$, then there are $I$ disjoint Golomb rulers, each being a $J$-subset of $A$, which generalizes the conjecture proposed by Koml{\'o}s, Sulyok and Szemer{\'e}di in 1975 on the special case $I = 1$. These conjectures are computationally verified for some values of $I$ and $J$ through modest computation. Eighteen exact values of $H(I,J)$ and ten upper bounds on $H(I,J)$ are obtained by computer search for $7 \leq I \leq 13$ and $10 \leq J \leq 13$. Moveover for $I > 13$ and $10 \leq J \leq 13$, $H(I,J)=IJ$ are determined without difficulty.