Even maps, the Colin de Verdiere number and representations of graphs

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10421445" target="_blank" >RIV/00216208:11320/20:10421445 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1137/1.9781611975994.161" target="_blank" >https://doi.org/10.1137/1.9781611975994.161</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/1.9781611975994.161" target="_blank" >10.1137/1.9781611975994.161</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Even maps, the Colin de Verdiere number and representations of graphs

Popis výsledku v původním jazyce

Van der Holst and Pendavingh introduced a graph parameter sigma, which coincides with the more famous Cohn de Verdiere graph parameter mu for small values. However, the definition of sigma is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured mu(G) &lt;= sigma(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on mu(G) which is, in general, tight. Equality between mu and sigma does not hold in general as van der Holst and Pendavingh showed that there is a graph G with mu(G) &lt;= 18 and sigma(G) &gt;= 20. We show that the gap appears on much smaller values, namely, we exhibit a graph H for which mu(H) &lt;= 7 and sigma(H) &gt;= 8. We also prove that, in general, the gap can be large: The incidence graphs H-q of finite projective planes of order q satisfy mu(H-q) is an element of O(q(3/2)) and sigma(H-q) &gt;= q(2).

Název v anglickém jazyce

Even maps, the Colin de Verdiere number and representations of graphs

Popis výsledku anglicky

Van der Holst and Pendavingh introduced a graph parameter sigma, which coincides with the more famous Cohn de Verdiere graph parameter mu for small values. However, the definition of sigma is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured mu(G) &lt;= sigma(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on mu(G) which is, in general, tight. Equality between mu and sigma does not hold in general as van der Holst and Pendavingh showed that there is a graph G with mu(G) &lt;= 18 and sigma(G) &gt;= 20. We show that the gap appears on much smaller values, namely, we exhibit a graph H for which mu(H) &lt;= 7 and sigma(H) &gt;= 8. We also prove that, in general, the gap can be large: The incidence graphs H-q of finite projective planes of order q satisfy mu(H-q) is an element of O(q(3/2)) and sigma(H-q) &gt;= q(2).

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-04113Y" target="_blank" >GJ19-04113Y: Pokročilé nástroje v kombinatorice, topologii a příbuzných oblastech</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

PROCEEDINGS OF THE THIRTY-FIRST ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA&apos;20)

ISBN

978-1-61197-599-4

ISSN

—

e-ISSN

—

Počet stran výsledku

16

Strana od-do

2642-2657

Název nakladatele

ASSOC COMPUTING MACHINERY

Místo vydání

NEW YORK

Místo konání akce

Salt Lake City

Datum konání akce

5. 1. 2020

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000554408102042