Circuits through cocircuits in a graph with extensions to matroids (Q2568499)

The circumference \(c(M)\) of a matroid \(M\) is defined to be the size of its largest circuit and the cocircumference \(c*(M)\) is the size of its largest cocircuit. Recently, Oxley posed the conjecture that for any connected matroid \(M\) with at least 2 elements, one can find a collection of at most \(c*(M)\) circuits which cover each element of \(M\) at least twice. Here, the author gives an affirmative answer to Oxley's conjecture for graphic matroids by proving the following Theorem 1.3. For a \(2\)-connected graph~\(G\), there is a collection of at most \(c*(G)\) cycles which cover each edge of \(G\) at least twice. Further, the author proves a corresponding result for cographic matroids, namely: Theorem 1.1. Let \(G\) be a \(k\)-connected graph where \(k\geq 2\). Then there is a cycle which intersects every bond of size \(c*-k+2\) or greater. Next, the author's theorem in [Combinatorica 25, 439--450 (2005; Zbl 1090.05038)] is extended to regular matroids. Namely, it is shown that for a \(k\)-connected regular matroid having circumference \(c\geq 2k\), if \(C_1\) and \(C_2\) are disjoint circuits satisfying \(r(C_1)+r(C_2)=r(C_1\cup C_2)\), then \(|C_1|+|C_2|\leq 2(c-k+1)\).