Logarithmic basis number of graphs and regular matroids

Abstract

The basis number $bn(G)$ of a graph $G$ is the minimum edge-congestion of a basis of its cycle space. We prove that every finite $n$-vertex multigraph satisfies $bn(G)=O(\log n)$, resolving, for simple graphs, a question of Bazargani, Biedl, Bose, Maheshwari and Miraftab, subsequently stated as a conjecture by Miraftab, Morin and Yuditsky. The argument also yields the cycle-rank refinement $bn(G)=O(\log \beta(G))$, where $\beta(G)$ is the dimension of the cycle space, and a reduction of Lehner and Miraftab, based on a theorem of Richter and Shank, then gives $bn(G)=O(\log g)$ for graphs of Euler genus $g$. For regular matroids we prove the ground-set bound $bn(M)=O(\log m)$, where $m=|E(M)|$, and logarithmic bounds in both the rank $r(M)$ and the cycle-space dimension $d$. All these orders are best possible.