Girth Conditions and Rota’s Basis Conjecture

Abstract

Rota’s basis conjecture (RBC) states that given a collection ${\mathcal {B}}$ of n bases in a matroid M of rank n, one can always find n disjoint rainbow bases with respect to ${\mathcal {B}}$. In this paper, we show that if M has girth at least $n-o(\sqrt{n})$, and no element of M belongs to more than $o(\sqrt{n})$ bases in ${\mathcal {B}}$, then one can find at least $n - o(n)$ disjoint rainbow bases with respect to ${\mathcal {B}}$. More specifically, we show that if M has girth at least $n- \beta (n) +1$ and each element belongs to no more than $\kappa (n)$ bases in ${\mathcal {B}}$, then letting $\gamma (n) = 4(\kappa (n) + \beta (n)+1)^2,$ one can find at least $n - \gamma (n)$ disjoint rainbow bases provided $2\gamma (n) < n$. This result can be seen as an extension of the work of Geelen and Humphries, who proved RBC in the case where M is paving, and ${\mathcal {B}}$ is a pairwise disjoint collection. The proofs here are based on modifications to the cascade idea introduced by Bucić et al.