منابع مشابه
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...
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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 → ∞. ...
متن کاملThe extremal function for K8- minors
We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computer-assisted.
متن کاملSmall Complete Minors Above the Extremal Edge Density
A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader’s result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest ...
متن کاملExtremal 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...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2001
ISSN: 0095-8956
DOI: 10.1006/jctb.2000.2013