On the Bounding Function of Cogirth and Supereulerian Regular Matroids

Following Bauer’s 1985 question on how a graph’s minimum degree relates to it being supereulerian, Catlin solved the problem for graphs with a minimum degree of at least n5. Subsequently, Lai solved it for the case where the minimum degree is at least n10. Projecting this problem into the setting of matroids leads to the question of the relationship between a matroid being supereulerian and its cogirth. In 2025, Qin et al. showed that a connected simple regular matroid M with cogirth at max{r(M)+115,9} is supereulerian. In this paper, we prove that for any real number c with 15<c<20, there exists a smallest integer f(c) such that every connected simple regular matroid M with cogirth at least max{r(M)+1c,f(c)} has a spanning Eulerian subset.