Cyclic Orders: Equivalence and Duality

András Sebő (1, 2) , Pierre Charbit (1, 2)

1 Leibniz - IMAG - Laboratoire Leibniz
2 UCBL - Université Claude Bernard Lyon 1

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomass\'{e}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 ``polar'' result of Bessy and Thomass\'{e}'s using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

András Sebő, Pierre Charbit. Cyclic Orders: Equivalence and Duality. 2006. ⟨hal-00005192⟩