an:06431540
Zbl 1311.05027
Alahmadi, A.; Aldred, R. E. L.; de la Cruz, R.; Ok, S.; Sol??, P.; Thomassen, C.
The minimum number of minimal codewords in an \([n, k]\)-code and in graphic codes
EN
Discrete Appl. Math. 184, 32-39 (2015).
00343528
2015
j
05B35 94B25
minimal codewords; matroid theory; cycle code of graphs
Summary: We survey some lower bounds on the function in the title based on matroid theory and address the following problem by \textit{G. Dosa} et al. [PU.M.A., Pure Math. Appl. 15, No. 4, 383--392 (2004; Zbl 1112.05021)]: Determine the smallest number of circuits in a loopless matroid with no parallel elements and with a given size and rank. In the graphic 3-connected case we provide a lower bound which is a product of a linear function of the number of vertices and an exponential function of the average degree. We also prove that, for \(p \geq 38\), every 3-connected graph with \(p\) vertices has at least as many cycles as the wheel with \(p\) vertices.
Zbl 1112.05021