Unavoidable Patterns in Complete Simple Topological Graphs
نویسندگان
چکیده
We show that every complete n-vertex simple topological graph contains a subgraph on at least $$(\log n)^{1/4 - o(1)}$$ vertices is weakly isomorphic to the convex geometric or twisted graph. This first improvement bound $$\varOmega (\log ^{1/8}n)$$ obtained in 2003 by Pach, Solymosi, and Tóth. also plane path of length n)^{1 -o(1)}$$ .
منابع مشابه
Unavoidable Configurations in Complete Topological Graphs
A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical...
متن کاملUnavoidable topological minors of infinite graphs
A graph G is loosely-c-connected, or l-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of l-c-connected infinite graphs that form an unavoidable set among the topological minors of l-c-connected infinite gr...
متن کاملSaturated simple topological graphs
A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with n vertices and O(n) edges. These constructions are nearly optimal: it is shown that ev...
متن کاملSaturated simple and k-simple topological graphs
A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with n vertices and O(n) edges. For every k > 1, we give similar constructions for k-simple...
متن کاملUnavoidable patterns
Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every 2 > 0 and positive integer k there is an n(k, 2) such that every 2-edge-coloring of the complete graph of order n ≥ n(k, 2) which has at least 2n2 ) edges in each color contains a member of Fk. This ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Lecture Notes in Computer Science
سال: 2023
ISSN: ['1611-3349', '0302-9743']
DOI: https://doi.org/10.1007/978-3-031-22203-0_1