Graphs and Combinatorics

, Volume 30, Issue 1, pp 171–181 | Cite as

Zero Divisors Among Digraphs

Original Paper

Abstract

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 \({A \times C \cong X \times C}\) .

Keywords

Digraphs Direct product of digraphs Cancellation 

Mathematics Subject Classification (2000)

05C76 

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Copyright information

© Springer Japan 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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