发表机构
Universitat de Barcelona; Centre de Recerca Matemàtica (CRM)（巴塞罗那大学; 数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明有限n顶点多重图的基数值bn(G)为O(log n)，还得出循环秩细化式、欧拉亏格为g的图的基数值阶，解决相关问题且这些阶最优。
AI 中文摘要
图G的基数值bn(G)是其循环空间基的最小边拥塞。我们证明，每个有限n顶点多重图都满足bn(G)=O(log n)，为简单图解决了Bazargani、Biedl、Bose、Maheshwari和Miraftab提出、后被Miraftab、Morin和Yuditsky表述为猜想的问题。该论证还得出循环秩细化式bn(G)=O(log β(G))，其中β(G)是循环空间的维数，结合Lehner和Miraftab基于Richter与Shank定理的归约，可得到欧拉亏格为g的图满足bn(G)=O(log g)，这些阶是最优的。
英文摘要
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 $bn(G)=O(\log β(G))$, where $β(G)$ 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.
Comments13 pages
