Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust (rainbow) subdivisions and simplicial cycles

István Tomon

Abstract

We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if $α>0$ and $\ell=Ω(\frac{1}α\log\frac{1}α)$ is an odd integer, then every graph $G$ with $n$ vertices and at least $n^{1+α}$ edges contains an $\ell$-subdivision of the complete graph $K_t$, where $t=n^{Θ(α)}$. Also, this remains true if in addition the edges of $G$ are properly colored, and one wants to find a rainbow copy of such a subdivision. In the sparser regime, we show that properly edge colored graphs on $n$ vertices with average degree $(\log n)^{2+o(1)}$ contain rainbow cycles, while average degree $(\log n)^{6+o(1)}$ guarantees rainbow subdivisions of $K_t$ for any fixed $t$, thus improving recent results of Janzer and Jiang et al., respectively. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if $G$ is a $2$-dimensional pure simplicial complex ($3$-graph) with $n$ $1$-dimensional and at least $n^{1+α}$ 2-dimensional faces, then $G$ contains a triangulation of the cylinder and the Möbius strip with $O(\frac{1}α\log\frac{1}α)$ vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well. In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs $G$ with strong expansion. We argue that if one randomly samples the vertices (and colors) of $G$ with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.

Robust (rainbow) subdivisions and simplicial cycles

Abstract

We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if and is an odd integer, then every graph with vertices and at least edges contains an -subdivision of the complete graph , where . Also, this remains true if in addition the edges of are properly colored, and one wants to find a rainbow copy of such a subdivision. In the sparser regime, we show that properly edge colored graphs on vertices with average degree contain rainbow cycles, while average degree guarantees rainbow subdivisions of for any fixed , thus improving recent results of Janzer and Jiang et al., respectively. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if is a -dimensional pure simplicial complex (-graph) with -dimensional and at least 2-dimensional faces, then contains a triangulation of the cylinder and the Möbius strip with vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well. In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs with strong expansion. We argue that if one randomly samples the vertices (and colors) of with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.
Paper Structure (17 sections, 35 theorems, 67 equations, 4 figures)

This paper contains 17 sections, 35 theorems, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Robust subdivisions
  3. Rainbow cycles and subdivisions
  4. Cycles in simplicial complexes
  5. Rainbow cycles and subdivisions
  6. Preliminaries
  7. a-maximal graphs
  8. Colorful expansion of random samples
  9. Proofs of the rainbow theorems
  10. Cycles in simplicial complexes
  11. Simplicial complexes and higher order walks
  12. a-maximal simplicial complexes
  13. Expansion in simplicial complexes
  14. Cycles in simplicial complexes --- Proof of Theorem \ref{['thm:cycleformal']}
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exist constants $c_1,c_2>0$ such that the following holds. Let $\alpha\in (0,1/2)$, $\ell\geq \frac{c_1}{\alpha}\log\frac{1}{\alpha}$ odd, and $n>n_0(\alpha,\ell)$. Let $G$ be a graph on $n$ vertices with at least $n^{1+\alpha}$ edges. Then $G$ contains the $\ell$-subdivision of $K_t$, where $

Figures (4)

  • Figure 1: A 3-uniform tight cycle of length 20 (left) and a tight cycle of length 21 (right).
  • Figure 2: An illustration of a path in a 3-graph, where the numbers denote the order of the vertices, and the red edges denote $f_0,\dots,f_{\ell}$.
  • Figure 3: Two cycles in 3-graphs. The left is homeomorphic to the cylinder, while the right is homeomorphic to the Möbius strip.
  • Figure 4: An illustration for the definition of $N(X,U)$ in 4-graphs.

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: Multiplicative Chernoff bound
  • Claim 2.2
  • Lemma 2.3
  • proof
  • ...and 63 more