Circuit Decompositions of Binary Matroids

.Given a simple Eulerian binary matroid \(M\), what is the minimum number of disjoint circuits necessary to decompose \(M\)? We prove that \({ }{M}{ }/ (rank(M)+1)\) many circuits suffice if \(M = F_2^n \{0\}\) is the complete binary matroid, for certain values of \(n\), and that \(O(2rank(M)/ (rank(M)+1))\) many circuits suffice for general \(M\). We also determine the asymptotic behavior of the minimum number of circuits in an odd-cover of \(M\).Keywordsbinary matroidmatroid circuitsarboricitycycle decompositionodd-coveringMSC codes05B3505C3805C70