TY - JOUR
T1 - Fast embedding of spanning trees in biased Maker-Breaker games
AU - Hefetz, Dan
AU - Ferber, Asaf
AU - Krivelevich, Michael
PY - 2011/12/1
Y1 - 2011/12/1
N2 - Given a tree T=(V, E) on n vertices, we consider the (1:q) Maker-Breaker tree embedding game Tn. The board of this game is the edge set of the complete graph on n vertices. Maker wins Tn if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α, ε>0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most nε, Maker has a winning strategy for the (1:q) game Tn, for every q≤nα. Moreover, we prove that Maker can win this game within n+o(n) moves which is clearly asymptotically optimal.
AB - Given a tree T=(V, E) on n vertices, we consider the (1:q) Maker-Breaker tree embedding game Tn. The board of this game is the edge set of the complete graph on n vertices. Maker wins Tn if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α, ε>0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most nε, Maker has a winning strategy for the (1:q) game Tn, for every q≤nα. Moreover, we prove that Maker can win this game within n+o(n) moves which is clearly asymptotically optimal.
KW - Embedding spanning trees
KW - Hamilton connected
KW - Maker-Breaker games
UR - http://www.scopus.com/inward/record.url?scp=82955216128&partnerID=8YFLogxK
U2 - 10.1016/j.endm.2011.09.054
DO - 10.1016/j.endm.2011.09.054
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AN - SCOPUS:82955216128
SN - 1571-0653
VL - 38
SP - 331
EP - 336
JO - Electronic Notes in Discrete Mathematics
JF - Electronic Notes in Discrete Mathematics
ER -