# 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

, Volume 88 (1) – Jan 1, 2018
20 pages

/lp/wiley/on-triangles-in-kr-minor-free-graphs-aQ5iIyOFVf
Publisher
Wiley Subscription Services, Inc., A Wiley Company
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: ; ; ;

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