Positional games on graphs

Abstract

<p>\section*{Abstract}<br />We study Maker-Breaker games played on the edges of the complete graph on $n$ vertices, $K_n$, whose family of winning sets $\cF$ consists of all edge sets of subgraphs $G\subseteq K_n$ which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming $a$, respectively $b$, unclaimed edges per move. We are interested in finding the threshold bias $b_{\cF}(a)$ for all values of $a$, so that for every $b$, $b\leq b_{\cF}(a)$, Maker wins the game and for all values of $b$, such that $b&gt;b_{\cF}(a)$, Breaker wins the game. We are particularly interested in cases where both $a$ and $b$ can be greater than $1$. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of $K_n$ and on the&nbsp; \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on $K_n$.<br /><br />Next, we consider biased $(1:b)$ Avoider-Enforcer games, also played on the edges of $K_n$. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries<br />to avoid claiming $k$ edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games, the strict and the monotone, and for each provide explicit winning strategies for both players. Consequentially, we establish bounds on the threshold biases $f^{mon}_\cF$, $f^-_\cF$ and $f^+_\cF$, where $\cF$ is the hypergraph of the game (the family of target sets).<br />We also study the monotone version of $K_{2,2}$-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to $K_{2,2}$ in $K_n$.&nbsp;&nbsp;</p><p>Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of $K_n$. Here, we look at the biased $(1:b)$ games, where Maker&#39;s bias is 1, and Breaker&#39;s bias is $b, b\ge 1$.</p>

<p>\section*{Izvod}</p><p>Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$, \v{c}ija familija pobedni\v{c}kih skupova $\cF$ obuhvata sve skupove grana grafa $G\subseteq K_n$ koji imaju neku monotono rastu\&#39;{c}u osobinu. Dva igra\v{c}a, \textit{Mejker} (\textit{Pravi\v{s}a}) i \textit{Brejker} (\textit{Kva\-ri\-\v{s}a}) se smenjuju u odabiru $a$, odnosno $b$, slobodnih grana po potezu. Interesuje nas da prona\dj emo grani\v{c}ni bias $b_{\cF}(a)$ za sve vrednosti pa\-ra\-me\-tra $a$, tako da za svako $b$, $b\le b_{\cF}(a)$, Mejker pobe\dj uje u igri, a za svako $b$, takvo da je $b&gt;b_{\cF}(a)$, Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra $a$ i $b$ mogu imati vrednost ve\&#39;cu od 1. Na\v{s}a pa\v{z}nja je posve\&#39;{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi&nbsp; grane svih pokrivaju\&#39;cih stabala grafa $K_n$, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa $K_n$.</p><p>Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom $(1:b)$, koje se tako\dj e igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$. Za svaku konstantu $k$, $k\ge 3$ analiziramo igru $k$-zvezde (zvezde sa $k$ krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima $k$ svojih grana incidentnih sa istim \v{c}vorom. Posmatramo obe verzije ove igre, striktnu i monotonu, i za svaku dajemo eksplicitnu pobedni\v{c}ku strategiju za oba igra\v{c}a. Kao rezultat, dobijamo gornje i donje ograni\v{c}enje za grani\v{c}ne biase $f^{mon}_\cF$, $f^-_\cF$ i $f^+_\cF$, gde $\cF$ predstavlja hipergraf igre (familija ciljnih skupova).<br />%$f^{mon}$, $f^-$ and $f^+$.<br />Tako\dj e, posmatramo i monotonu verziju $K_{2,2}$-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa $K_{2,2}$.</p><p>Kona\v{c}no, \v{z}elimo da prona\dj emo strategije za brzu pobedu Mejkera u igrama savr\v{s}enog me\v{c}inga i Hamiltonove konture, koje se tako\dj e igraju na granama kompletnog grafa $K_n$. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera $b$, $b\ge 1$.</p>