GRAPH SATURATION GAMES Ago-Erik Riet 1 joint work with Jonathan Lee Estonian Theory Days of Computer Science, Jõeküla October 2015 1 ago-erik.riet@ut.ee - My work was partially supported by Foundation Archimedes and European Social Fund, and by the Estonian Research Council through the research grants PUT405, PUT620, IUT20-57.
Combinatorial games with winners • Combinatorial game theory studies, among others, the following kind of games which have a winner. • Strong positional game: there is a game board and some subsets of the game board are winning. Players claim the elements (squares) of the game board in turn. Examples include tic-tac-toe – with winning sets three in a row, gomoku – five in a row – etc. The game board could be the edge set of a graph. • Maker-Breaker game: one player has to claim a winning set. The other player wins if this never happens before the whole board is claimed. • Avoider-Enforcer game: one player wants to force the other one to claim a losing set. • However, some games have no winner but a score instead.
Extremal number • What is the maximal number of edges in a graph on n vertices if it has no triangle? • The answer is given by Mantel’s Theorem (1907). The only extremal graph is u ⌊ n / 2 ⌋ u 1 u 2 . . . v 1 v 2 v 3 v ⌈ n / 2 ⌉ . . .
Saturation number • What is the minimal number of edges on n vertices if • it does not contain a triangle • but adding any edge creates a triangle? • Clearly the extremal graph is connected. • But it could already be a connected graph of minimal size, more precisely a star: u v n − 1 v 1 v 2 v 3 . . .
Extremal number and saturation number • So the extremal number (Turán number) or maximal number of edges for a triangle ex ( n , K 3 ) is ⌊ n 2 4 ⌋ . • and the saturation number or minimal number of edges for a triangle Sat ( n , K 3 ) is n − 1.
Game saturation number • A graph is triangle-saturated if it is maximal triangle-free. • What is the “usual” number of edges in a saturated graph? • Idea: Füredi, Reimer ja Seress (1991) considered the following game. • Two players Prolonger and Shortener build a triangle-saturated graph on n vertices adding edges alternately, starting with the empty graph. • Their goal is to maximize, respectively minimize the final number of edges, called the score. • The total number of edges at the end when the graph has become triangle-saturated on optimal play by both players is called the score or the game saturation number G ( K 3 ) .
The game saturation number for the triangle • Füredi, Reimer and Seress proved that Prolonger can guarantee score ≥ 1 2 n log 2 n − o ( n log n ) . • A very short proof sketch: • Prolonger builds a matching with ⌊ n 2 ⌋ edges. v 1 v 2 ... v r ... u 1 u 2 u r • In such graphs the number of edges is at least that large.
Game saturation number for the triangle • Füredi, Reimer and Seress also spread the rumor that there is a lost proof by Erdős that Shortener can guarantee that the final score is ≤ n 2 5 . • Bíró, Horn and Wildstrom thought the idea was to cover almost all vertices by disjoint 5-cycles - but it seems Prolonger can counteract that. In a triangle-free graph, the maximal number of edges between two 5-cycles is 10. • Bíró, Horn and Wildstrom proved that Shortener can guarantee that the final score is ≤ 26 121 n 2 + o ( n 2 ) .
Our results • We looked at similar games — but avoiding another graph instead of a triangle: • Let P k be the path on k vertices. We looked at the P 4 and P 5 avoidance games ( P 5 illustrated) • This game for general P k where Prolonger is allowed to skip moves. • Also the game where players build a digraph, avoiding a directed walk on k vertices. • And some more games...
Our results: bounds for the game saturation number graph or graphs lower bound upper bound 1 1 P k , Prolonger 4 n ( k − 2 ) 2 n ( k − 1 ) can pass 5 n − 14 4 4 P 4 5 n + 1 5 P 5 n − 1 n + 2 T k Write n = q ( k − 1 ) + r : Write n = q ( k − 1 ) + r : � k − 1 � r � k − 1 � r � � � � q + − ( k − 3 ) q + 2 2 2 2 1 1 K 1 , k + 1 2 ( kn − 2 ( k − 1 )) 2 kn for n ≥ 3 k 2 − 3 k − 4 3 n 2 + 1 3 n 2 + 1 1 3 nk + O ( n + k 2 ) 1 3 nk + O ( n + k 2 ) directed walk P k , k ≥ 4 � n � 1 � n � 1 ( a , b ) -biased, ( 1 − k λ − )( 1 + o ( 1 )) ( 1 − k λ + )( 1 + o ( 1 )) 2 2 where λ − = where λ + = ⌊ b / 2 a ⌋ 1 directable P k + 1 1 + ⌊ b / 2 a ⌋ 1 + ⌊ a / 2 b ⌋
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 v a 2 a 4 a 1 Figure : Unifying a Hamiltonian component with an isolated vertex.
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 v a 2 S a 4 a 1 Figure : Unifying a Hamiltonian component with an isolated vertex.
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 P v a 2 S a 4 a 1 Figure : Unifying a Hamiltonian component with an isolated vertex.
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 b 2 b 3 a 2 b 4 a 4 b 1 a 1 Figure : Unifying two Hamiltonian components
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 b 2 b 3 a 2 b 4 a 4 b 1 S a 1 Figure : Unifying two Hamiltonian components
Avoiding the path P k : Prolonger’s strategy if he is allowed to skip moves. a 3 a 5 P b 2 b 3 a 2 b 4 a 4 b 1 S a 1 Figure : Unifying two Hamiltonian components
Types of components in a P 4 -saturated graph. Figure : Types of components in a P 4 -saturated graph.
Types of components in a P 5 -saturated graph. Figure : Types of components in a P 5 -saturated graph.
Types of components in a P 6 -saturated graph. Figure : Types of components in a P 6 -saturated graph.
Shortener’s strategy at the game avoiding the walk P k . v k − 4 v k − 3 v k − 2 v k − 1 v 1 v 2 v 3 v 4 u 1 u 2 Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.
Shortener’s strategy at the game avoiding the walk P k . v k − 4 v k − 3 v k − 2 v k − 1 v 1 v 2 v 3 v 4 u 1 u 2 Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.
Shortener’s strategy at the game avoiding the walk P k . v k − 4 v k − 3 v k − 2 v k − 1 v 1 v 2 v 3 v 4 u 1 u 2 u 3 Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.
Shortener’s strategy at the game avoiding the walk P k . v k − 4 v k − 3 v k − 2 v k − 1 v 1 v 2 v 3 v 4 u 1 u 2 u 3 Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.
Shortener’s strategy at the game avoiding the walk P k . v k − 4 v k − 3 v k − 2 v k − 1 v 1 v 2 v 3 v 4 u 1 u 2 u 3 Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.
Prolonger’s strategy at the game avoiding the walk P k . λ + 1 λ Figure : Structure A λ . Prolonger’s strategy: force all vertices into structures of large λ .
Prolonger’s strategy at the game avoiding the walk P k . λ + 1 λ − 1 Figure : Structure B λ . Prolonger’s strategy: force all vertices into structures of large λ .
Prolonger’s strategy at the game avoiding the walk P k . λ + 1 λ − 2 Figure : Structure C λ . Prolonger’s strategy: force all vertices into structures of large λ .
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