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 (K-6 and K-7, 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 K-8-minor or contains K-2,K- 2,K- 2,K- 2,K- 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 K-7-minor free graph is 8-colorable and every K-8-minor free graph is 10-colorable.