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Manipulative Waiters with Probabilistic Intuition

Published online by Cambridge University Press:  21 December 2015

MAŁGORZATA BEDNARSKA-BZDȨGA
Affiliation:
Faculty of Mathematics and CS, Adam Mickiewicz University, Poznań, Poland (e-mail: mbed@amu.edu.pl, tomasz@amu.edu.pl)
DAN HEFETZ
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: danny.hefetz@gmail.com)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, 6997801, Israel (e-mail: krivelev@post.tau.ac.il)
TOMASZ ŁUCZAK
Affiliation:
Faculty of Mathematics and CS, Adam Mickiewicz University, Poznań, Poland (e-mail: mbed@amu.edu.pl, tomasz@amu.edu.pl)

Abstract

For positive integers n and q and a monotone graph property $\mathcal{A}$ , we consider the two-player, perfect information game WC(n, q, $\mathcal{A}$ ), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, q + 1 edges of the complete graph Kn which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining q edges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\mathcal{A}$ , then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter q is close to n and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if q ⩾ (1 + ϵ)n, then Client can avoid components of order cϵ−2 ln n for some absolute constant c > 0, whereas for q ⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for every qcn, where c > 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.

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Copyright © Cambridge University Press 2015 

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