Logarithmic basis number of graphs

Knauer proves that every finite $n$-vertex multigraph satisfies $\mathrm{bn}(G)=O(\log n)$, improving the previous general $O(\log^2 n)$ bound and matching the known $\Omega(\log n)$ order. He also proves the sharper cycle-rank bound $\mathrm{bn}(G)=O(\log\beta(G))$. Combined with a reduction of Lehner and Miraftab, this yields $\mathrm{bn}(G)=O(\log g)$ for graphs of Euler genus $g$, improving the previous $O(\log^2 g)$ bound to the optimal logarithmic order.

Knauer proves that every finite $n$-vertex multigraph satisfies $\mathrm{bn}(G)=O(\log n)$, improving the previous general $O(\log^2 n)$ bound and matching the known $\Omega(\log n)$ order. He also proves the sharper cycle-rank bound $\mathrm{bn}(G)=O(\log\beta(G))$. Combined with a reduction of Lehner and Miraftab, this yields $\mathrm{bn}(G)=O(\log g)$ for graphs of Euler genus $g$, improving the previous $O(\log^2 g)$ bound to the optimal logarithmic order.