Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $ntimes n$ array $A$ based on ${0,1,cdots,nd}$ is called emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of ${1,2,cdots,nd}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $nequiv1,5pmod 6$, and it is proved that for any $nequiv1,5pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2leq dleq n-1$.
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