Constructions of regular sparse anti-magic squares
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For positive integers $n$ and $d$ with $d<n$, an $n\times n$ array $A$ based on $\mathcal{X}=\{0,1,\ldots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each non-zero element of $\mathcal{X}$ occurs exactly once in $A$, 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 $d$ non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1$, $5\pmod 6$, and prove that there exists a regular SAMS$(n,d)$ for any $n\geq 5$, $n\equiv1$, $5\pmod 6$ and $d$ with $2\leq d\leq n-1$.
Abstract
For positive integers $n$ and $d$ with $d<n$, an $n\times n$ array $A$ based on $\mathcal{X}=\{0,1,\ldots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each non-zero element of $\mathcal{X}$ occurs exactly once in $A$, 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 $d$ non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1$, $5\pmod 6$, and prove that there exists a regular SAMS$(n,d)$ for any $n\geq 5$, $n\equiv1$, $5\pmod 6$ and $d$ with $2\leq d\leq n-1$.
참고문헌 (24)
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