We make the first step towards a "nerve theorem" for graphs. Let G be a simple graph and let F be a family of induced subgraphs of G such that the intersection of any members of F is either empty or connected. We show that if the nerve complex of F has non-vanishing homology in dimension three, then G contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar (p, q) theorem due to Alon and Kleitman: Let F be a finite family of open connected sets in the plane such that the intersection of any members of F is either empty or connected. If among any p >= 3 members of F there are some three that intersect, then there is a set of C points which intersects every member of F, where C is a constant depending only on p.

- Publisher
- AMER MATHEMATICAL SOC

- Issue Date
- 2019-06

- Language
- English

- Article Type
- Article

- Citation
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.371, no.12, pp.8755 - 8779

- ISSN
- 0002-9947

- Appears in Collection
- MA-Journal Papers(저널논문)

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