On a conjecture of Gross, Mansour and Tucker

Sergei Chmutov (1) , Fabien Vignes-Tourneret (2)

1 Mathematics Department, The Ohio State University
2 PSPM - Probabilités, statistique, physique mathématique

Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler-Poincaré duality. This operation often changes the genus. Recently J. L. Gross, T. Mansour, and T. W. Tucker formulated a conjecture that for any ribbon graph different from plane trees and their partial duals, there is a subset of edges partial duality relative to which does change the genus. A family of counterexamples was found by Qi Yan and Xian'an Jin. In this note we prove that essentially these are the only counterexamples.

Mots clés: Ribbon graphs; partial duality; partial-dual genus polynomial; Gross–Mansour–Tucker conjecture

Domaines: Combinatoire [math.CO]; Topologie géométrique [math.GT]

Sergei Chmutov, Fabien Vignes-Tourneret. On a conjecture of Gross, Mansour and Tucker. European Journal of Combinatorics, 2021, 97, pp.103368. ⟨10.1016/j.ejc.2021.103368⟩. ⟨hal-03121632v2⟩