Cyclic orders: Equivalence and duality

Abstract

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent.

We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.

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Correspondence to Pierre Charbit.

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Charbit, P., Sebő, A. Cyclic orders: Equivalence and duality. Combinatorica 28, 131 (2008). https://doi.org/10.1007/s00493-008-2197-0

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C20
  • 05C70
  • 90C10
  • 90C27