The 3-connected binary matroids with circumference 6 or 7
For a matroid \(M\) that has a circuit, \(\text{circ}(M)\) denotes the circumference of \(M\), that is, the maximum cardinality of a circuit of \(M\). Recently, the circumference of a matroid has appeared in some bounds; [see \textit{M. Lemos} and \textit{J. Oxley}, Discrete Math. 189, No.\,1-3, 283-289 (1998; Zbl 0956.05029); J. Graph Theory 30, No.\,1, 51--66 (1999; Zbl 0933.05086); Discrete Math. 220, No.\,1-3, 145--157 (2000; Zbl 0951.05020)]. The results of this paper are used in \textit{R. Cordovil, B. M. Junior,} and \textit{M. Lemos} [Discrete Math. 309, No.\,4, 655--665 (2009; Zbl 1169.05012)] to improve a lower bound of \textit{M. Lemos} and \textit{J. Oxley} [Discrete Math. 220, No.,1-3, 145--157 (2000; Zbl 0951.05020] for the size of a 3-connected binary matroid having a circuit whose deletion originates also a 3-connected matroid. In this paper, the authors construct all 3-connected binary matroids with circumference equal to 6 or 7 having large rank. Moreover, they conjecture that the following problems are very difficult: {\parindent=5mm \begin{itemize}\item[1.]constructing all 3-connected matroids with circumference 6 or 7 (and large rank), \item[2.]constructing all the 3-connected binary matroids with circumference 8. \end{itemize}}
scientific article; zbMATH DE number 5641706