2025/02/26 by Csikvári, Péter
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.19196
The Merino-Welsh conjecture states that for a graph G without loops and bridges we have max(TG(2,0),TG(0,2))≥ TG(1,1). Later Jackson proved that for any matroid M without loop and coloop we have TM(3,0)TM(0,3)≥ TM(1,1)2. The value 3 in this statement was improved to 2.9242 by Beke, Csáji, Csikvári and Pituk. In this paper, we further improve on this result by showing that TM(2.355,0)TM(0,2.355)≥ TM(1,1)2. We also prove that the Merino--Welsh conjecture is true for matroids M, where all circuits of M and its dual M^* have length between ℓ and (ℓ-2)4 for some ℓ≥ 6.