On Crossing Numbers of Complete Tripartite and Balanced Complete Multipartite Graphs
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The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.
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The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.
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1909-present
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- Department of Electrical Engineering (1909-1985)
- Department of Electrical Engineering and Computer Engineering (1985-1995)
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- College of Engineering (parent college)
- Department of Physics and Electrical Engineering (predecessor)
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Abstract
The crossing number cr(G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number (cr) over bar (G) of G is the minimum number of crossings in a such drawing of G with edges as straight line segments. Zarankiewicz proved in 1952 that (cr) over bar (K-n1,K- n2)
A(n1, n2, n3) :=
[GRAPHICS]
(left perpendicular n(j)/2 right perpendicular left perpendicular n(j)-1/2 right perpendicular left perpendicular n(k)/2 right perpendicular left perpendicular n(k)-1/2 right perpendicular + left perpendicular n(i)/2 right perpendicular left perpendicular n(i)-1/2 right perpendicular left perpendicular n(j)n(k)/2 right perpendicular),
and prove (cr) over bar (K-n1,K- n2,K- n3) infinity of cr(K-n,K- n) over the maximum number of crossings in a drawing of K-n,K- n exists and is at most 1/4. We define zeta(r) := 3(r(2) - r)/8(r(2) + r-3) and show that for a fixed r and the balanced complete r- partite graph, zeta(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.
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This is the peer-reviewed version of the following article: Gethner, Ellen, Leslie Hogben, Bernard Lidický, Florian Pfender, Amanda Ruiz, and Michael Young. "On crossing numbers of complete tripartite and balanced complete multipartite graphs." Journal of Graph Theory 84, no. 4 (2017): 552-565, which has been published in final form at DOI: 10.1002/jgt.22041. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.