Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs

Behague, Natalie C.; Marbach, Trent G.; Prałat, Paweł; Ruciński, Andrzej

doi:10.48550/arxiv.2105.07034

“…Definitions. In this paper, we consider the semi-random graph process suggested by Peleg Michaeli, introduced formally in [5], and studied recently in [3,4,8,11,13,14,15,22] that can be viewed as a "one player game". The process starts from G 0 , the empty graph on the vertex set [n] := {1, .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, it was conjectured that part (b) can be generalized to such large family of graphs. The conjecture was proved recently in [3]. As a result, creating graphs of a constant size is well-understood-essentially, creating a fixed graph with degeneracy d is possible once the process lasts long enough so that there are vertices with at least d − 1 squares.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Gamarnik¹,

Kang²,

Prałat³

2023

Preprint

1

0

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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.

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“…Definitions. In this paper, we consider the semi-random graph process suggested by Peleg Michaeli, introduced formally in [3], and studied recently in [2,8,9,1] that can be viewed as a "one player game". The process starts from G 0 , the empty graph on the vertex set [n] := {1, .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The program is available on-line. 1 Moreover, by investigating some specific structures that are generated by the semi-random process, which guarantee the existence of a large set of families of edges that cannot simultaneously contribute to the construction of a Hamiltonian cycle, we improve the lower bound of ln 2 + ln(1 + ln 2) ≥ 1.21973 to 1.26575. The structures we investigate in this work are different from the ones in [8].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

Gao¹,

MacRury²,

Prałat³

2022

Preprint

Self Cite

0

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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.We focus on the problem of constructing a Hamiltonian cycle in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves it in αn rounds, where α < 2.01678 is derived from the solution to some system of differential equations. We also show that the player cannot achieve the desired property in less than βn rounds, where β > 1.26575. These results improve the previously best known bounds and, as a result, the gap between the upper and lower bounds is decreased from 1.39162 to 0.75102.

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Sharp thresholds in adaptive random graph processes

MacRury,

Surya

2023

Random Struct Algorithms

1

0

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The ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).

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Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs

Cited by 4 publications

References 0 publications

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

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

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

Sharp thresholds in adaptive random graph processes

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