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The $k$-representation number of the random graph

Ayush Basu, Vojtěch Rödl, Marcelo Sales

Abstract

The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In particular, for $k=1$ this notion is equivalent to the clique number of a graph $G$. Extending results of Frieze and Reed, and Eaton and Grable, we study the $k$-representation number of $G(n,1/2)$. As a tool, we will prove a sharp concentration result counting the number of induced subgraphs of $G(n,1/2)$ with density $(\frac{1}{2}+α)$. In Lemma 3.7, we will show that the number of such subgraphs is close to its expected value with probability $1-\exp(-n^C)$.

The $k$-representation number of the random graph

Abstract

The -representation number of a graph is the minimum cardinality of the system of vertex subsets with the property that every edge of is covered at least times while every non-edge is covered at most times. In particular, for this notion is equivalent to the clique number of a graph . Extending results of Frieze and Reed, and Eaton and Grable, we study the -representation number of . As a tool, we will prove a sharp concentration result counting the number of induced subgraphs of with density . In Lemma 3.7, we will show that the number of such subgraphs is close to its expected value with probability .
Paper Structure (25 sections, 15 theorems, 223 equations, 1 figure)

This paper contains 25 sections, 15 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivalent Statement, Notation and Concentration Inequalities
  3. Equivalence with the condition of $k$-covering
  4. Notation:
  5. Concentration Inequalities
  6. Proof of Theorem \ref{['thm:main']}
  7. Lower Bound:
  8. Upper Bound:
  9. Proof of Lower Bound
  10. Definition of Fractional Pseudocover.
  11. Proof of Lemma \ref{['thm: Lemma3.2']}
  12. Proof of Upper Bound
  13. Proof of Counting Lemma
  14. Organisation of Proof:
  15. Extension Property
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exist absolute positive constants $c_1, c_2 > 0$ such that for all $\varepsilon > 0$, there exists $n_1=n_1(\varepsilon)$, such that if $n\geq n_1$ and $1< k< n^{\frac{1}{2}-\varepsilon}$, with high probability and if $(\log \log n)^{1/\varepsilon}\leq k\leq \log n$, with high probability.

Figures (1)

  • Figure 1: Choices for $z_{i+1}$ in the random process

Theorems & Definitions (63)

  • Theorem 1.1
  • Definition 2.1: $k$-cover
  • proof
  • Lemma 2.3: Chernoff Bounds
  • Lemma 2.4: Azuma Hoeffding Inequality
  • Definition 3.1: Property $\mathscr{P}_1$
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6: $(\alpha,t)$-good graphs
  • ...and 53 more