On triangles in Kr‐minor free graphs

On triangles in Kr‐minor free graphs We study graphs where each edge that is incident to a vertex of small degree (of degree at most 7 and 9, respectively) belongs to many triangles (at least 4 and 5, respectively) and show that these graphs contain a complete graph (K6 and K7, respectively) as a minor. The second case settles a problem of Nevo. Moreover, if each edge of a graph belongs to six triangles, then the graph contains a K8‐minor or contains K2, 2, 2, 2, 2 as an induced subgraph. We then show applications of these structural properties to stress freeness and coloring of graphs. In particular, motivated by Hadwiger's conjecture, we prove that every K7‐minor free graph is 8‐colorable and every K8‐minor free graph is 10‐colorable. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Graph Theory Wiley

On triangles in Kr‐minor free graphs

Loading next page...
 
/lp/wiley/on-triangles-in-kr-minor-free-graphs-aQ5iIyOFVf
Publisher
Wiley Subscription Services, Inc., A Wiley Company
Copyright
Copyright © 2018 Wiley Periodicals, Inc.
ISSN
0364-9024
eISSN
1097-0118
D.O.I.
10.1002/jgt.22203
Publisher site
See Article on Publisher Site

Abstract

We study graphs where each edge that is incident to a vertex of small degree (of degree at most 7 and 9, respectively) belongs to many triangles (at least 4 and 5, respectively) and show that these graphs contain a complete graph (K6 and K7, respectively) as a minor. The second case settles a problem of Nevo. Moreover, if each edge of a graph belongs to six triangles, then the graph contains a K8‐minor or contains K2, 2, 2, 2, 2 as an induced subgraph. We then show applications of these structural properties to stress freeness and coloring of graphs. In particular, motivated by Hadwiger's conjecture, we prove that every K7‐minor free graph is 8‐colorable and every K8‐minor free graph is 10‐colorable.

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 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

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