Year: 2024
Type: preprint
Abstract: The basis number of a graph $G$ is the smallest integer $k$ such that $G$ admits a basis $B$ for its cycle space, where each edge of $G$ belongs to at most $k$ members of $B$. In this note, we show that every non-planar graph that can be embedded on a surface with Euler characteristic $0$ has a basis number of exactly $3$, proving a conjecture of Schmeichel from 1981. Additionally, we show that any graph embedded on a surface $Σ$ (whether orientable or non-orientable) of genus $g$ has a basis number of $O(\log^2 g)$.
Source: arXiv (Cornell University)
Authors: Lehner, Florian, Miraftab, Babak
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403571077