Extremal theory for convex matchings in convex geometric graphs

Abstract

A convex geometric graph G of order n consists of the set of vertices of a plane convex n-gon P together with some edges and/or diagonals of P as edges. Call G l-free if G does not have l disjoint edges in convex position. We answer the following questions: (a) What is the maximum possible number of edges of G if G is l-free (as a function of n and 1)? (b) What is the minimum possible number of edges of G if G is l-free and saturated, i.e., if G U {e} is not l-free for any edge or diagonal e of P that is not already in G. We also fully describe the graphs G where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word "disjoint" from the definition of "l-free" and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turk's theorem, etc.

Extremal problems for convex geometric hypergraphs and ordered hypergraphs

“…Given a cgg F, let ex ↻ (n, F) denote the maximum number of edges in an n-vertex cgg that does not contain F as a cgg (defined analogously to the linearly ordered case). Extremal problems for geometric graphs have a long history, beginning with theorems on disjoint line segments [11,13,17], to more recent results on crossing matchings [3,5]. In the vein of Conjecture A, Braß [2] asked for the determination of all acyclic graphs F such that ex ↻ (n, F) is linear in n, and this problem remains open (recently it was solved for trees [9]).…”

Geometric Graphs with No Three Disjoint Edges

“…There are many nice results for various forbidden classes-k pairwise crossing edges, k pairwise "parallel" edges, k pairwise disjoint edges, self-crossing paths, even cycles and many others (see [1], [2], [7], [8], [11], and [15]- [17]). For a survey of results on geometric graphs see [10].…”

Tight paths in convex geometric hypergraphs

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In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12], Sutherland [19], Kupitz and Perles [16] for convex geometric graphs, as well as the classical Erdős-Gallai Theorem [6] for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.

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