On the connectivity Waiter-Client game

• Main Authors: Antoniuk, Sylwia, Grosu, Codruut, Narins, Lothar
• Format: Journal Article
• Language: English
• Published: 20-10-2015
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.1510.05852

In this short note we consider a variation of the connectivity Waiter-Client
game $WC(n,q,\mathcal{A})$ played on an $n$-vertex graph $G$ which consists of
$q+1$ disjoint spanning trees. In this game in each round Waiter offers Client
$q+1$ edges of $G$ which have not yet been offered. Client chooses one edge and
the remaining $q$ edges are discarded. The aim of Waiter is to force Client to
build a connected graph. If this happens Waiter wins. Otherwise Client is the
winner. We consider the case where $2 < q+1 < \lfloor \frac{n-1}{2}\rfloor$ and
show that for each such $q$ there exists a graph $G$ for which Client has a
winning strategy. This result stands in opposition to the case where $G$
consists of just 2 spanning trees or where $G$ is a complete graph, since it
has been shown that for such graphs Waiter can always force Client to build a
connected graph.