Improvements on the density of maximal 1‐planar graphs

Improvements on the density of maximal 1‐planar graphs A graph is 1‐planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1‐planar drawing is called 1‐plane. A graph is maximal 1‐planar (1‐plane), if we cannot add any missing edge so that the resulting graph is still 1‐planar (1‐plane). Brandenburg et al. showed that there are maximal 1‐planar graphs with only 4517n+O(1)≈2.647n edges and maximal 1‐plane graphs with only 73n+O(1)≈2.33n edges. On the other hand, they showed that a maximal 1‐planar graph has at least 2813n−O(1)≈2.15n−O(1) edges, and a maximal 1‐plane graph has at least 2.1n−O(1) edges. We improve both lower bounds to 20n9≈2.22n. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Graph Theory Wiley

Improvements on the density of maximal 1‐planar graphs

Loading next page...
 
/lp/wiley/improvements-on-the-density-of-maximal-1-planar-graphs-qY1xGwJl2N
Publisher
Wiley
Copyright
Copyright © 2018 Wiley Periodicals, Inc.
ISSN
0364-9024
eISSN
1097-0118
D.O.I.
10.1002/jgt.22187
Publisher site
See Article on Publisher Site

Abstract

A graph is 1‐planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1‐planar drawing is called 1‐plane. A graph is maximal 1‐planar (1‐plane), if we cannot add any missing edge so that the resulting graph is still 1‐planar (1‐plane). Brandenburg et al. showed that there are maximal 1‐planar graphs with only 4517n+O(1)≈2.647n edges and maximal 1‐plane graphs with only 73n+O(1)≈2.33n edges. On the other hand, they showed that a maximal 1‐planar graph has at least 2813n−O(1)≈2.15n−O(1) edges, and a maximal 1‐plane graph has at least 2.1n−O(1) edges. We improve both lower bounds to 20n9≈2.22n.

Journal

Journal of Graph TheoryWiley

Published: Jan 1, 2018

Keywords: ;

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off