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%2F22%3A10476095" target="_blank" >RIV/00216208:11320/22:10476095 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=n1-X1aHe-f" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=n1-X1aHe-f</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00493-021-4443-7" target="_blank" >10.1007/s00493-021-4443-7</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 Colin de Verdiere graph parameter mu for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured sigma(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 sigma(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 at much smaller values, namely, we exhibit a graph H for which mu(H) &gt;= 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 Colin de Verdiere graph parameter mu for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured sigma(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 sigma(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 at much smaller values, namely, we exhibit a graph H for which mu(H) &gt;= 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

J_imp - Článek v periodiku v databázi Web of Science

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

2022

Kód důvěrnosti údajů

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

Název periodika

Combinatorica

ISSN

0209-9683

e-ISSN

1439-6912

Svazek periodika

42

Číslo periodika v rámci svazku

SUPPL 2 / Supplement 2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

29

Strana od-do

1317-1345

Kód UT WoS článku

000798210100003

EID výsledku v databázi Scopus

2-s2.0-85130418329