kConnectedness and kFactors in the SemiRandom Graph Process
Abstract
The semirandom graph process is a singleplayer graph game where the player is initially
presented an edgeless graph with n vertices. In each round, the player is offered a vertex
u uniformly at random and subsequently chooses a second vertex v deterministically ac
cording to some strategy, and adds edge uv to the graph. The objective for the player is
then to ensure that the graph fulfils some specified property as fast as possible.
We investigate the properties of being kconnected and containing a kfactor. We settle
the open case for 2connectedness by showing that the player has a strategy to construct a
2connected graph asymptotically almost surely in (ln 2+ln(ln 2+1)+o(1))n rounds, which
matches a known lower bound asymptotically. We also provide a strategy for building a
kfactor asymptotically almost surely in (β + 10^−5)n rounds, where β is derived from the
solution of a system of differential equations.
Additionally, we consider a variant that was recently suggested by Wormald where the
player chooses the first vertex and the second vertex is chosen uniformly at random. We
show that the bounds for kconnectedness for the traditional setting are also tight for this
variant.
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Cite this version of the work
Hidde Koerts
(2022).
kConnectedness and kFactors in the SemiRandom Graph Process. UWSpace.
http://hdl.handle.net/10012/18976
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