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Cliques, Chromatic Number, and Independent Sets in the Semi-random Process

David Gamarnik, Mihyun Kang, Pawel Pralat

TL;DR

This work analyzes the semi-random graph process as a one-player game where a random vertex is paired with a strategically chosen partner to rapidly enforce monotone properties. The authors develop a framework combining concentration bounds and the differential equation method to obtain near-tight bounds on three targets: the largest achievable clique size $K_k$, the chromatic number $ ext{χ}(G_t)$, and the independence number $ ext{α}(G_t)$, across multiple time regimes $t$ relative to $n$. They present explicit lower- and upper-bound results in regimes such as $t=n^{1+o(1)}$, $t=eta n ext{ log }n$, and $t o ext{ω}n ext{ log }n$, linking $ ext{χ}$ and the clique number and providing near-optimal control of $ ext{α}(G_t)$ via degenerate-structure arguments and edge-removal analyses. The findings advance understanding of how adaptivity interacts with randomness to shape clique formation, colorability, and independence in semi-random graphs, with implications for related stochastic processes and graph construction problems.

Abstract

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order $k$, having the chromatic number at least $k$, and not having an independent set of size at least $k$.

Cliques, Chromatic Number, and Independent Sets in the Semi-random Process

TL;DR

This work analyzes the semi-random graph process as a one-player game where a random vertex is paired with a strategically chosen partner to rapidly enforce monotone properties. The authors develop a framework combining concentration bounds and the differential equation method to obtain near-tight bounds on three targets: the largest achievable clique size , the chromatic number , and the independence number , across multiple time regimes relative to . They present explicit lower- and upper-bound results in regimes such as , , and , linking and the clique number and providing near-optimal control of via degenerate-structure arguments and edge-removal analyses. The findings advance understanding of how adaptivity interacts with randomness to shape clique formation, colorability, and independence in semi-random graphs, with implications for related stochastic processes and graph construction problems.

Abstract

The semi-random graph process is a single player game in which the player is initially presented an empty graph on vertices. In each round, a vertex is presented to the player independently and uniformly at random. The player then adaptively selects a vertex , and adds the edge to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order , having the chromatic number at least , and not having an independent set of size at least .
Paper Structure (24 sections, 14 theorems, 58 equations, 3 figures)

This paper contains 24 sections, 14 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction and Main Results
  2. Definitions
  3. Notation
  4. Main Results---Complete Graphs
  5. Main Results---Chromatic Number
  6. Main Results---Independent Sets
  7. Structure of the Paper
  8. Preliminaries
  9. Concentration Tools
  10. The Differential Equation Method
  11. Literature Review
  12. Perfect Matchings
  13. Hamilton Cycles
  14. Spanning Subgraphs
  15. A Few Other Directions
  16. ...and 9 more sections

Key Result

Theorem 1.3

Suppose that $t = t(n)$ is such that $t=n^{1+o(1)}$ and $t \ll n \log n$. Let $\beta = \beta(n) \ :=\ n \log n / t$. (In particular, $\beta \to \infty$, as $n \to \infty$.) Define and (In particular, $\epsilon = o(1)$, regardless of $\beta$.) Finally, let Then, the following hold.

Figures (3)

  • Figure 1: The upper ($k'$) and the lower ($k$) bound for the order of a largest complete graph: small (left figure) and large (right figure) values of $\gamma$.
  • Figure 2: The ratio between the upper ($k'$) and the lower ($k$) bound for the order of a largest complete graph: small (left figure) and large (right figure) values of $\gamma$.
  • Figure 3: The upper ($2\ell+2$) and the lower ($k$) bound for $\chi(G_t)$ (left figure) as well as the ratio between the two (right figure).

Theorems & Definitions (23)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:small_t']}(a) and the first part of Theorem \ref{['thm:large_t']}(a)
  • ...and 13 more