---
res:
bibo_abstract:
- "Van der Holst and Pendavingh introduced a graph parameter σ, which coincides
with the more famous Colin de Verdière graph parameter μ for small values. However,
the definition of a is much more geometric/topological directly reflecting embeddability
properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G)
for any graph G. We confirm this conjecture. As far as we know, this is the first
topological upper bound on σ(G) which is, in general, tight.\r\nEquality between
μ and σ does not hold in general as van der Holst and Pendavingh showed that there
is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much
smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8.
We also prove that, in general, the gap can be large: The incidence graphs Hq
of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Vojtech
foaf_name: Kaluza, Vojtech
foaf_surname: Kaluza
foaf_workInfoHomepage: http://www.librecat.org/personId=21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E
orcid: 0000-0002-2512-8698
- foaf_Person:
foaf_givenName: Martin
foaf_name: Tancer, Martin
foaf_surname: Tancer
foaf_workInfoHomepage: http://www.librecat.org/personId=38AC689C-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1191-6714
bibo_doi: 10.1007/s00493-021-4443-7
bibo_volume: 42
dct_date: 2022^xs_gYear
dct_identifier:
- UT:000798210100003
dct_isPartOf:
- http://id.crossref.org/issn/0209-9683
dct_language: eng
dct_publisher: Springer Nature@
dct_title: Even maps, the Colin de Verdière number and representations of graphs@
...