Received: 29 January 2014 Revised: 21 June 2017 Accepted: 18 July 2017
On triangles in
-minor free graphs
Boris Albar Daniel Gonçalves
LIRMM, CNRS and University of Montpellier,
Daniel Gonçalves, LIRMM, CNRS and
University of Montpellier 2, 161 rue
Ada, 34095 Montpellier cedex5, France.
We study graphs where each edge that is incident to a ver-
tex 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 (
, 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
-minor or con-
as an induced subgraph. We then show appli-
cations of these structural properties to stress freeness and
coloring of graphs. In particular, motivated by Hadwiger's
conjecture, we prove that every
-minor free graph is 8-
colorable and every
-minor free graph is 10-colorable.
coloration, graph, minors, stress freeness
A minor of a graph is a graph obtained from by a succession of edge deletions, edge contractions
and vertex deletions. All graphs we consider are simple, that is, without loops or multiple edges. The
following theorem of Mader  bounds the number of edges in a
-minor free graph.
Theorem 1.1. For 3 ≤ ≤ 7,any
-minor free graph on ≥ vertices has at most ( −2) −
Note that since () =
deg(), this theorem implies that, for 3 ≤ ≤ 7,every
minor free graph has minimum degree () ≤ 2 −5. This property will be of importance in the
following. We are interested in a suﬃcient condition for a graph to admit a complete graph as a minor,
based on the minimum number of triangles each edge belongs to. Nevo  already studied this prob-
lem for small cliques. In the following, we assume that every graph has at least one edge.
Theorem 1.2. For 3 ≤ ≤ 5,any
-minor free graph has an edge that belongs to at most −3
154 © 2017 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/jgt J Graph Theory. 2018;88:154–173.