  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.
