Extremal functions for rooted minors

Extremal Functions for Graph Linkages and Rooted Minors

Interval minors of complete bipartite graphs

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in Kr,s-interval minor free bipartite graphs. We determine exact values when r = 2 and describe the extremal graphs. For r = 3, lower and upper bounds are given and the structure of K3,s-interval minor free graphs is studied.

Bounds on Minors of Binary Matrices

We prove an upper bound on sums of squares of minors of {+1, −1}-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘(1, −1)-matrices with near-extremal properties’, SIAM J. Discrete Math. 23 (2009), 1422–1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices. 2010 Mathematics subject classification: primary 05B...

The Extremal Functions for Triangle-free Graphs with Excluded Minors

We prove two results: 1. A graph G on at least seven vertices with a vertex v such that G − v is planar and t triangles satisfies |E(G)| ≤ 3|V (G)| − 9 + t/3. 2. For p = 2, 3, . . . , 9, a triangle-free graph G on at least 2p − 5 vertices with no Kp-minor satisfies |E(G)| ≤ (p− 2)|V (G)| − (p− 2).

Graph Minors

For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b + 1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n/ b+f b such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several e...