TY - JOUR

T1 - Very fast construction of bounded-degree spanning graphs via the semi-random graph process

AU - Ben-Eliezer, Omri

AU - Gishboliner, Lior

AU - Hefetz, Dan

AU - Krivelevich, Michael

N1 - Publisher Copyright:
© 2020 Wiley Periodicals LLC

PY - 2020/12

Y1 - 2020/12

N2 - In this paper, we study the following recently proposed semi-random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded-degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n-vertex graph with maximum degree (Formula presented.) can be constructed with high probability in (Formula presented.) rounds. This is tight up to a multiplicative factor of (Formula presented.). We also obtain tight bounds for the number of rounds necessary to embed bounded-degree spanning trees, and consider a nonadaptive variant of this setting.

AB - In this paper, we study the following recently proposed semi-random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded-degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n-vertex graph with maximum degree (Formula presented.) can be constructed with high probability in (Formula presented.) rounds. This is tight up to a multiplicative factor of (Formula presented.). We also obtain tight bounds for the number of rounds necessary to embed bounded-degree spanning trees, and consider a nonadaptive variant of this setting.

KW - embedding spanning graphs

KW - semi-random graph process

UR - http://www.scopus.com/inward/record.url?scp=85091504764&partnerID=8YFLogxK

U2 - 10.1002/rsa.20963

DO - 10.1002/rsa.20963

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AN - SCOPUS:85091504764

SN - 1042-9832

VL - 57

SP - 892

EP - 919

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 4

ER -