The Kőnig graph process

Nina Kamčev; Michael Krivelevich; Natasha Morrison; Benny Sudakov

doi:10.1002/rsa.20969

The Kőnig graph process

Nina Kamčev, Michael Krivelevich, Natasha Morrison, Benny Sudakov

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Say that a graph G has property (Formula presented.) if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set (Formula presented.) and let e₁, e₂, … e_N be a uniformly random ordering of the edges of K_n, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge e_{m + 1} exactly if Gm ∪ {e_{m + 1}} has property (Formula presented.). We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?.

Original language	English
Pages (from-to)	1272-1302
Number of pages	31
Journal	Random Structures and Algorithms
Volume	57
Issue number	4
DOIs	https://doi.org/10.1002/rsa.20969
State	Published - 1 Dec 2020

Keywords

matching
perfect matching
random graph
random process
vertex cover

All Science Journal Classification (ASJC) codes

Software
General Mathematics
Computer Graphics and Computer-Aided Design
Applied Mathematics

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10.1002/rsa.20969

Cite this

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title = "The K{\H o}nig graph process",

abstract = "Say that a graph G has property (Formula presented.) if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set (Formula presented.) and let e1, e2, … eN be a uniformly random ordering of the edges of Kn, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge em + 1 exactly if Gm ∪ {em + 1} has property (Formula presented.). We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?.",

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AU - Krivelevich, Michael

AU - Morrison, Natasha

AU - Sudakov, Benny

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N2 - Say that a graph G has property (Formula presented.) if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set (Formula presented.) and let e1, e2, … eN be a uniformly random ordering of the edges of Kn, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge em + 1 exactly if Gm ∪ {em + 1} has property (Formula presented.). We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?.

AB - Say that a graph G has property (Formula presented.) if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set (Formula presented.) and let e1, e2, … eN be a uniformly random ordering of the edges of Kn, with n an even integer. Let G0 be the empty graph on n vertices. For m ≥ 0, Gm + 1 is obtained from Gm by adding the edge em + 1 exactly if Gm ∪ {em + 1} has property (Formula presented.). We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of GN and for which m will Gm contain a perfect matching?.

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JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 4

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The Kőnig graph process

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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