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Zero Divisors Among Digraphs


A digraph C is called a zero divisor if there exist non-isomorphic digraphs A and B for which \({A \times C \cong B \times C}\) , where the operation is the direct product. In other words, C being a zero divisor means that cancellation property \({A \times C \cong B \times C \Rightarrow A \cong B}\) fails. Lovász proved that C is a zero divisor if and only if it admits a homomorphism into a disjoint union of directed cycles of prime lengths.Thus any digraph C that is homomorphically equivalent to a directed cycle (or path) is a zero divisor. Given such a zero divisor C and an arbitrary digraph A, we present a method of computing all solutions X to the digraph equation \({A \times C \cong X \times C}\) .

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Correspondence to Richard Hammack.

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This work extends and generalizes some earlier results by R. Hammack and K. Toman [Cancellation of direct products of digraphs, Discusiones Mathematicae Graph Theory, 30 (2010) 575–590].

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Hammack, R., Smith, H. Zero Divisors Among Digraphs. Graphs and Combinatorics 30, 171–181 (2014).

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  • Digraphs
  • Direct product of digraphs
  • Cancellation

Mathematics Subject Classification (2000)

  • 05C76