منابع مشابه
Minors in expanding graphs
In this paper we address several extremal problems related to graph minors. In all of our results we assume essentially that a given graph G is expanding, where expansion is either postulated directly, or G can be shown to contain a large expanding subgraph, or G is locally expanding due to the fact that G does not contain a copy of a fixed bipartite graph H. We need the following definitions t...
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Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|, |Y | ≥ 7 there are at least six edges between X and Y . We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either • G \ v is planar for some vertex v, or • G can be drawn with crossings in the plane, but wi...
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Let be a real number such that 0< <2 and t a positive integer. Let n be a sufficiently large positive integer as a function of t and . We show that every n-vertex graph with at least n1+ edges contains a subdivision of Kt in which each edge of Kt is subdivided less than 10/ times. This refines the main result in [A. Kostochka and Pyber, Combinatorica 8 (1988), 83–86] and resolves an open questi...
متن کاملMinors in random regular graphs
We show that there is a constant c so that for fixed r ≥ 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c √ n vertices as a minor. This confirms a conjecture of Markström [17]. Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of...
متن کاملMinors in lifts of graphs
We study here lifts and random lifts of graphs, as defined in [1]. We consider the Hadwiger number η and the Hajós number σ of `lifts of Kn, and analyze their extremal as well as their typical values (that is, for random lifts). When ` = 2, we show that n2 ≤ η ≤ n, and random lifts achieve the lower bound (as n → ∞). For bigger values of `, we show Ω ( n √ logn ) ≤ η ≤ n √ `. We do not know how...
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ژورنال
عنوان ژورنال: Geometric and Functional Analysis
سال: 2009
ISSN: 1016-443X,1420-8970
DOI: 10.1007/s00039-009-0713-z