Switchings, extensions, and reductions in central digraphs

André Kündgen, Gregor Leander, Carsten Thomassen

Journal of Combinatorial Theory, Series A Volume 118, Issue 7, October 2011, Pages 2025–2034


Abstract

A directed graph is called central if its adjacency matrix A satisfies the equation A2=J, where J is the matrix with a 1 in each entry. It has been conjectured that every central directed graph can be obtained from a standard example by a sequence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We disprove these conjectures and present a general extension result which, in particular, shows that each counterexample extends to an infinite family.

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Tags: Central, directed, graph;, groupoid;, path, property, unique