Supereulerian regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29-44] that for sufficiently large n $n$, every 2-edge-connected simple graph G $G$ with n = divide V( G ) divide $n=| V(G)| $ and minimum degree delta( G ) >= n 5 $\delta (G)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765-1776], it is shown that for any connected simple regular matroid M $M$, if every cocircuit D $D$ of M $M$ satisfies divide D divide >= max r( M ) - 5 5 , 6 $| D| \ge \max \left\{\frac{r(M)-5}{5},6\right\}$, then M $M$ is supereulerian. We prove the following. (i) Let M $M$ be a connected simple regular matroid. If every cocircuit D $D$ of M $M$ satisfies divide D divide >= max r( M ) + 1 10 , 9 $| D| \ge \max \left\{\frac{r(M)+1}{10},9\right\}$, then M $M$ is supereulerian. (ii) For any real number c $c$ with 0 < c < 1 $0\lt c\lt 1$, there exists an integer f( c ) $f(c)$ such that if every cocircuit D $D$ of a connected simple cographic matroid M $M$ satisfies divide D divide >= max{c(r( M ) + 1 ) , f( c ) } $| D| \ge \max \{c(r(M)+1),f(c)\}$, then M $M$ is supereulerian.