GRAPH SATURATION GAMES Ago-Erik Riet 1 joint work with Jonathan Lee - - PowerPoint PPT Presentation

GRAPH SATURATION GAMES

Ago-Erik Riet1 joint work with Jonathan Lee Estonian Theory Days of Computer Science, Jõeküla

October 2015

1ago-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
• ther 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
• nly extremal graph is

v1 v2 v3 . . . v⌈n/2⌉ u1 u2 . . . u⌊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:

v1 v2 v3 . . . vn−1 u

Extremal number and saturation number

• So the extremal number (Turán number) or maximal

number of edges for a triangle ex(n, K3) is ⌊ n2

4 ⌋.

• and the saturation number or minimal number of edges

for a triangle Sat(n, K3) 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(K3).

The game saturation number for the triangle

• Füredi, Reimer and Seress proved that Prolonger can

guarantee score ≥ 1

2n log2 n − o(n log n).

• A very short proof sketch:
• Prolonger builds a matching with ⌊ n

2⌋ edges.

u1 u2 ... ur v1 v2 ... vr

• 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 ≤ n2 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 121n2 + o(n2).

Our results

• We looked at similar games — but avoiding another

graph instead of a triangle:

• Let Pk be the path on k vertices. We looked at the P4

and P5 avoidance games (P5 illustrated)

• This game for general Pk 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 Pk, Prolonger

1 4n(k − 2) 1 2n(k − 1)

can pass P4

4 5n − 14 5 4 5n + 1

P5 n − 1 n + 2 Tk Write n = q(k − 1) + r: Write n = q(k − 1) + r: q k−1

2

• +

r

2

• − (k − 3)

q k−1

2

• +

r

2

• K1,k+1

1 2(kn − 2(k − 1)) 1 2kn

for n ≥ 3k2 − 3k − 4 directed walk Pk, k ≥ 4

1 3n2 + 1 3nk + O(n + k2) 1 3n2 + 1 3nk + O(n + k2)

(a, b)-biased, n

2

• (1 −

1 kλ− )(1 + o(1))

n

2

• (1 −

1 kλ+ )(1 + o(1))

directable Pk+1 where λ− =

⌊b/2a⌋ 1+⌊b/2a⌋

where λ+ =

1 1+⌊a/2b⌋

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 v

Figure : Unifying a Hamiltonian component with an isolated vertex.

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 v S

Figure : Unifying a Hamiltonian component with an isolated vertex.

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 v S P

Figure : Unifying a Hamiltonian component with an isolated vertex.

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 b1 b2 b3 b4

Figure : Unifying two Hamiltonian components

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 b1 b2 b3 b4 S

Figure : Unifying two Hamiltonian components

Avoiding the path Pk: Prolonger’s strategy if he is allowed to skip moves.

a1 a2 a3 a4 a5 b1 b2 b3 b4 S P

Figure : Unifying two Hamiltonian components

Types of components in a P4-saturated graph.

Figure : Types of components in a P4-saturated graph.

Types of components in a P5-saturated graph.

Figure : Types of components in a P5-saturated graph.

Types of components in a P6-saturated graph.

Figure : Types of components in a P6-saturated graph.

Shortener’s strategy at the game avoiding the walk Pk. v1 v2 v3 v4 vk−4 vk−3 vk−2 vk−1 u1 u2

Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.

Shortener’s strategy at the game avoiding the walk Pk. v1 v2 v3 v4 vk−4 vk−3 vk−2 vk−1 u1 u2

Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.

Shortener’s strategy at the game avoiding the walk Pk. v1 v2 v3 v4 vk−4 vk−3 vk−2 vk−1 u1 u2 u3

Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.

Shortener’s strategy at the game avoiding the walk Pk. v1 v2 v3 v4 vk−4 vk−3 vk−2 vk−1 u1 u2 u3

Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.

Shortener’s strategy at the game avoiding the walk Pk. v1 v2 v3 v4 vk−4 vk−3 vk−2 vk−1 u1 u2 u3

Figure : Shortener’s strategy: force into 3 classes the vertices of 3-vertex paths.

Prolonger’s strategy at the game avoiding the walk Pk.

λ + 1 λ

Figure : Structure Aλ. Prolonger’s strategy: force all vertices into structures of large λ.

Prolonger’s strategy at the game avoiding the walk Pk.

λ + 1 λ − 1

Figure : Structure Bλ. Prolonger’s strategy: force all vertices into structures of large λ.

Prolonger’s strategy at the game avoiding the walk Pk.

λ + 1 λ − 2

Figure : Structure Cλ. Prolonger’s strategy: force all vertices into structures of large λ.