6 research outputs found
The biased odd cycle game
In this paper we consider biased Maker-Breaker games played on the edge set
of a given graph . We prove that for every and large enough ,
there exists a constant for which if and
, then Maker can build an odd cycle in the game for
. We also consider the analogous game where Maker and
Breaker claim vertices instead of edges. This is a special case of the
following well known and notoriously difficult problem due to Duffus, {\L}uczak
and R\"{o}dl: is it true that for any positive constants and , there
exists an integer such that for every graph , if , then
Maker can build a graph which is not -colorable, in the
Maker-Breaker game played on the vertices of ?Comment: 10 page
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Pozicione igre na grafovima
\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on vertices, , whose family of winning sets \cF consists of all edge sets of subgraphs which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming , respectively , unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of , so that for every , b\leq b_{\cF}(a), Maker wins the game and for all values of , such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both and can be greater than . We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of and on the \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on . Next, we consider biased Avoider-Enforcer games, also played on the edges of . For every constant we analyse the -star game, where Avoider tries to avoid claiming 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). We also study the monotone version of -game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to in . Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of . Here, we look at the biased games, where Maker's bias is 1, and Breaker's bias is .\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa \v{c}vorova, , \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa koji imaju neku monotono rastu\'{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 , odnosno , 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 , tako da za svako , b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako , takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra i mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi grane svih pokrivaju\'cih stabala grafa , kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa . Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom , koje se tako\dj e igraju na granama kompletnog grafa sa \v{c}vorova, . Za svaku konstantu , analiziramo igru -zvezde (zvezde sa krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima 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). %, and . Tako\dj e, posmatramo i monotonu verziju -igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa . 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 . Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera ,
Pozicione igre na grafovima
\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on vertices, , whose family of winning sets \cF consists of all edge sets of subgraphs which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming , respectively , unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of , so that for every , b\leq b_{\cF}(a), Maker wins the game and for all values of , such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both and can be greater than . We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of and on the \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on . Next, we consider biased Avoider-Enforcer games, also played on the edges of . For every constant we analyse the -star game, where Avoider tries to avoid claiming 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). We also study the monotone version of -game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to in . Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of . Here, we look at the biased games, where Maker's bias is 1, and Breaker's bias is .\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa \v{c}vorova, , \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa koji imaju neku monotono rastu\'{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 , odnosno , 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 , tako da za svako , b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako , takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra i mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi grane svih pokrivaju\'cih stabala grafa , kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa . Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom , koje se tako\dj e igraju na granama kompletnog grafa sa \v{c}vorova, . Za svaku konstantu , analiziramo igru -zvezde (zvezde sa krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima 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). %, and . Tako\dj e, posmatramo i monotonu verziju -igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa . 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 . Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera ,
Pozicione igre na grafovima
\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on vertices, , whose family of winning sets \cF consists of all edge sets of subgraphs which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming , respectively , unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of , so that for every , b\leq b_{\cF}(a), Maker wins the game and for all values of , such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both and can be greater than . We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of and on the \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on . Next, we consider biased Avoider-Enforcer games, also played on the edges of . For every constant we analyse the -star game, where Avoider tries to avoid claiming 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). We also study the monotone version of -game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to in . Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of . Here, we look at the biased games, where Maker's bias is 1, and Breaker's bias is .\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa \v{c}vorova, , \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa koji imaju neku monotono rastu\'{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 , odnosno , 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 , tako da za svako , b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako , takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra i mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi grane svih pokrivaju\'cih stabala grafa , kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa . Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom , koje se tako\dj e igraju na granama kompletnog grafa sa \v{c}vorova, . Za svaku konstantu , analiziramo igru -zvezde (zvezde sa krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima 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). %, and . Tako\dj e, posmatramo i monotonu verziju -igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa . 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 . Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera ,