Infinite families of crossing-critical graphs with prescribed average degree and crossing number

Abstract

Širan constructed infinite families of ▫$k$▫-crossing-critical graphs for every ▫$k \ge 3$▫ and Kochol constructed such families of simple graphs for every ▫$k \ge 2$▫. Richter and Thomassen argued that, for any given ▫$k \ge 1$▫ and ▫$r \ge 6$▫, there are only finitely many simple ▫$k$▫-crossing-critical graphs with minimum degree ▫$r$▫. Salazar observed that the same argument implies such a conclusion for simple ▫$k$▫-crossing-critical graphs of prescribed average degree ▫$r > 6$▫. He established existence of infinite families of simple ▫$k$▫-crossing-critical graphs with any prescribed rational average degree ▫$r \in [4,6)$▫ for infinitely many ▫$k$▫ and asked about their existence for ▫$r \in (3,4)$▫. The question was partially settled by Pinontoan and Richter, who answered it positively for ▫$r \in (3\frac{1}{2},4)$▫. The present contribution uses two new constructions of crossing critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar's question by the following statement: for every rational number ▫$r \in (3,6)$▫ there exists an integer ▫$N_r$▫, such that, for any ▫$k > N_r$▫, there exists an infinite family of simple 3-connected crossing-critical graphs with average degree ▫$r$▫ and crossing number ▫$k$▫. Moreover, a universal lower bound on ▫$k$▫ applies for rational numbers in any closed interval ▫$I \subset (3,6)$▫.

Keywords

matematika;teorija grafov;prekrižno število;kritičen graf;prekrižno-kritičen graf;povprečna stopnja;mathematics;graph theory;crossing number;critical graph;crossing-critical graph;average degree;graph;

Data

Language:	English
Year of publishing:	2010
Typology:	1.01 - Original Scientific Article
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	519.17
COBISS:	15438169
ISSN:	0364-9024
Views:	48
Downloads:	18
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary keywords:	matematika;teorija grafov;prekrižno število;kritičen graf;prekrižno-kritičen graf;povprečna stopnja;
Type (COBISS):	Not categorized
Pages:	str. 139-162
Volume:	ǂVol. ǂ65
Issue:	ǂiss. ǂ2
Chronology:	2010
DOI:	10.1002/jgt.20470
ID:	1474841