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Constructions of regular sparse anti-magic squares
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 and , an
array based on is called \emph{a sparse anti-magic
square of order with density }, denoted by SAMS, if each element
of occurs exactly one entry of , and its row-sums,
column-sums and two main diagonal sums constitute a set of consecutive
integers. An SAMS is called \emph{regular} if there are exactly
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 , and it is proved that for any ,
there exists a regular SAMS if and only if .Comment: 18 page