On triangles in ‐minor free graphs

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.

The extremal function for K9= minors

“…Since G is 6-connected by (9), N C ( ) 1 and N C ( ) 2 each contain at least two nonadjacent pairs of vertices of N x ( ). Thus it is possible to pick distinct pairs from each of N C ( ) 1 and N C ( ) 2 , say v w N C , ( )…”

“…Now given two H H k ( , , ) 1 2 -cockades G 1 and G 2 , we let G be the graph obtained from G 1 and G 2 by identifying a clique of size k in G 1 with a clique of size k in G 2 . Then G is an H H k ( , , ) 1 2 -cockade, and every H H k ( , , ) 1 2…”

“…-cockade can be constructed in this fashion. If H 1 is isomorphic to H 2 , we simply write H k ( , ) 1 -cockade. Jørgensen [10] showed that any graph on n 8…”

“…Note that there are two nonisomorphic graphs K t = . Let K t = (1) denote the graph obtained from K t by deleting two incident edges, and let K t =(2) denote the graph obtained from K t by deleting two independent edges. Let…”

Graphs with no K9= minor are 10‐colorable

“…Graphs with no minor were first shown to be ‐colorable by Jakobsen in an unpublished paper. Graphs with no minor for were shown to be ‐colorable by Albar and Gonçalves with computer assistance. The present author and Song extended this last result to the case , and provided an alternative, computer‐free proof for the cases .…”