Crossing number additivity over edge cuts

Bokal, Drago; Chimani, Markus; Leanõs, Jesús

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Title:	Crossing number additivity over edge cuts
Authors:	ID Bokal, Drago (Author) ID Chimani, Markus (Author) ID Leanõs, Jesús (Author)
Files:	http://dx.doi.org/10.1016/j.ejc.2013.02.002
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Consider a graph $G$ with a minimal edge cut $F$ and let $G_1$ , $G_2$ be the two (augmented) components of $G-F$ . A long-open question asks under which conditions the crossing number of $G$ is (greater than or) equal to the sum ofcthe crossing numbers of $G_1$ and $G_2$ - which would allow us to consider those graphs separately. It is known that crossing number is additive for $\|F\| in {0,1,2}$ and that there exist graphs violating this property with $\|F\| ge 4$ . In this paper, we show that crossing number is additive for $\|F\|=3$ , thus closing the final gap in the question. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing-critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.
Keywords:	matematika, teorija grafov, prekrižno število, minor, mathematics, graph theory, crossing number, minor
Year of publishing:	2013
Number of pages:	str. 1010-1018
Numbering:	Vol. 34, iss. 6
PID:	20.500.12556/DKUM-52013
UDC:	519.17
ISSN on article:	0195-6698
COBISS.SI-ID:	16624473
NUK URN:	URN:SI:UM:DK:CCTIUJTW
Publication date in DKUM:	10.07.2015
Views:	1385
Downloads:	98
Metadata:
Categories:	Misc.
:	BOKAL, Drago, CHIMANI, Markus and LEANÕS, Jesús, 2013, Crossing number additivity over edge cuts. European journal of combinatorics [online]. 2013. Vol. 34, no. 6, p. 1010–1018. [Accessed 22 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.ejc.2013.02.002 Copy citation