Matroids with the circuit cover property
Let \(M\) be a binary matroid and \(p: M \mapsto Z^+\) be a weight on \(M\). A family \(\mathcal F\) of circuits of \(M\) is called a faithful circuit cover of \(M\) with respect to \(p\) if every element \(e\) of \(M\) is contained in precisely \(p(e)\) members of \(\mathcal F\). The following condition for \(p\), called the admissibility condition, is an obvious necessary condition for a matroid to have a faithful cover with respect to \(p\): (1) for every cocircuit \(D\) of \(M\), the total weight of \(D\) must be even; (2) for every cocircuit \(D\) and every \(e \in D\), \(p(e) \leq p(D)/2\). The faithful circuit cover theorem by \textit{B. Alspach, L. Goddyn} and \textit{C.-Q. Zhang} [Trans. Am. Math. Soc. 344, No. 1, 131-154 (1994; Zbl 0810.05043)] for graphs is generalized in this paper to binary matroids. That is, a matroid \(M\) has a faithful circuit cover with respect every admissible weight \(p\) if it does not have a minor isomorphic to any of \(F_7^*\), \(R_{10}\), \(M^*(K_5)\) or \(M(P_{10})\).
scientific article; zbMATH DE number 1251349