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 of Graph Theory – Wiley
Published: Jan 1, 2018
Keywords: ; ; ;
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