Logarithmic basis number of graphs

2026-09-02

AI Summary

本文解决了关于图的基数问题，证明了每个有限n顶点多重图的基数是O(log n)，并提供了基于循环空间维度和其他相关参数的新界限。

Abstract

The basis number $\mathrm{bn}(G)$ 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 \[ \mathrm{bn}(G)=O(\log n), \] 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 \[ \mathrm{bn}(G)=O(\log β(G)), \] 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 \[ \mathrm{bn}(G)=O(\log g) \] for graphs of Euler genus $g$. These orders are best possible.

Problem

• basis number
• logarithmic bound
• cycle space
• Euler genus

Innovation

• logarithmic basis number
• cycle space
• Euler genus
• edge-congestion