Extremal functions for sparse minors

Extremal functions for rooted minors

The graph G contains a graph H as a minor if there exist pair-wise disjoint sets {Si ⊆ V (G)|i = 1, . . . , |V (H)|} such that for every i, G[Si] is a connected subgraph and for every edge uv in H, there exists an edge of G with one end in Su and the other end in Sv. A rooted H minor in G is a minor where each Si of minor contains a predetermined xi ∈ V (G). We prove that if the constant c is s...

Extremal Functions for Graph Linkages and Rooted Minors

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).

The Extremal Function for Complete Minors

Let G and H be graphs. As usual, we say that H is a minor or subcontraction of G if V(G) contains disjoint subsets Wu , u # V(H), such that G[Wu] is connected for each u # V(H) and there is an edge in G between Wu and Wv whenever uv # E(H). (Here, G[Wu] stands for the subgraph of G induced by Wu ; our notation is standard and follows that of Bolloba s [2].) We write GoH if H is a minor of G, an...

The Extremal Function For Noncomplete Minors

We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let c(H) = inf { c : e(G) ≥ c|G| implies G ≻ H } . We define a parameter γ(H) of the graph H and show that, if H has t vertices, then c(H) = (αγ(H) + o(1)) t √ log t where α = 0.319 . . . is an explicit constant and o(1) denotes a term tending to zero as t → ∞. ...