A Tutte polynomial inequality for lattice path matroids

Kolja Knauer (1, 2) , Leonardo Martínez-Sandoval (3, 4) , Jorge Luis Ramírez Alfonsín (3)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille
2 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
3 IMAG - Institut Montpelliérain Alexander Grothendieck
4 Instituto de Matematicas [México]

Let M be a matroid without loops or coloops and let T (M ; x, y) be its Tutte polynomial. In 1999 Merino and Welsh conjectured that max(T (M ; 2, 0), T (M ; 0, 2)) ≥ T (M ; 1, 1) holds for graphic matroids. Ten years later, Conde and Merino proposed a multi-plicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.

Mots clés: Merino-Welsh conjecture; Tutte polynomial; lattice path matroids

Domaines: Informatique [cs]

Kolja Knauer, Leonardo Martínez-Sandoval, Jorge Luis Ramírez Alfonsín. A Tutte polynomial inequality for lattice path matroids. Advances in Applied Mathematics, 2016, 94, pp.23-38. ⟨10.1016/j.aam.2016.11.008⟩. ⟨hal-01457675⟩