Basis Number of Graphs Excluding Minors
Colin Geniet, Ugo Giocanti

TL;DR
This paper proves that graphs excluding a fixed minor have bounded basis number, extending classical results and using advanced graph decomposition techniques, with implications for understanding cycle structures in complex graphs.
Contribution
It establishes that graphs excluding a fixed minor have bounded basis number, generalizing Mac Lane's theorem and employing the Graph Minor Structure Theorem and tree-decomposition analysis.
Findings
Graphs of treewidth k have basis number bounded by a function of k.
Graphs excluding a fixed minor have basis number bounded by a polynomial in |H|.
The proof framework adapts techniques from Courcelle's conjecture and related results.
Abstract
The basis number of a graph is the minimum such that the cycle space of is generated by a family of cycles using each edge at most times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis number at most 2, and more generally, graphs embedded on a fixed surface of bounded genus are known to have bounded basis number. Generalising this, we prove that graphs excluding a fixed minor have bounded basis number. Our proof uses the Graph Minor Structure Theorem, which requires us to understand how basis number behaves in tree-decompositions. In particular, we prove that graphs of treewidth have basis number bounded by some function of . We handle tree-decompositions using the proof framework developed by Boja\'nczyk and Pilipczuk in their proof of Courcelle's conjecture. Combining our approach with independent results of Miraftab,…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
