Two Vertices Left Game
Tyler Seacrest
University of Nebraska–Lincoln
May 13, 2011
Coprime Game
David Ginat in Winning Moves and Illuminating Mathematical
Patterns introduced the following game.
Coprime Game
David Ginat in Winning Moves and Illuminating Mathematical
Patterns introduced the following game.
Game
A sequence of an even amount of consecutive integers is
written down.
Coprime Game
David Ginat in Winning Moves and Illuminating Mathematical
Patterns introduced the following game.
Game
A sequence of an even amount of consecutive integers is
written down.
Alice and Bob take turns crossing out numbers, with Alice
moving first, until there are two numbers left.
Coprime Game
David Ginat in Winning Moves and Illuminating Mathematical
Patterns introduced the following game.
Game
A sequence of an even amount of consecutive integers is
written down.
Alice and Bob take turns crossing out numbers, with Alice
moving first, until there are two numbers left.
If they have a common factor other than 1, then Alice
wins, and if they are coprime, then Bob wins.
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Bob
Coprime Game
Alice: Common Factor
Bob: Coprime
Example
Alice
Alice wins!
Bob
Coprime Game Solution
Coprime Game Solution
Bob has a winning strategy
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob
Coprime Game Solution
Bob has a winning strategy
Key Fact: Consecutive integers are coprime!
Example
Alice
Bob wins!
Bob
Posá’s Soup Problem
Over soup, Erdős posed to 12-year-old Posá:
Posá’s Soup Problem
Over soup, Erdős posed to 12-year-old Posá:
É
Prove that given n + 1 positive integers all less than or
equal to 2n, there exists a pair that are relatively prime.
Posá’s Soup Problem
Over soup, Erdős posed to 12-year-old Posá:
É
Prove that given n + 1 positive integers all less than or
equal to 2n, there exists a pair that are relatively prime.
Posá’s response:
Posá’s Soup Problem
Over soup, Erdős posed to 12-year-old Posá:
É
Prove that given n + 1 positive integers all less than or
equal to 2n, there exists a pair that are relatively prime.
Posá’s response:
É
“The two are consecutive."
Posá’s Soup Problem
Over soup, Erdős posed to 12-year-old Posá:
É
Prove that given n + 1 positive integers all less than or
equal to 2n, there exists a pair that are relatively prime.
Posá’s response:
É
“The two are consecutive."
Erdős: “Champagne would have been more appropriate
than soup."
Coprime Game and the Factor Graph
Definition
A factor graph is a graph with a consecutive set of integers for vertices, where i
and j are adjacent if they share a common
factor, other than 1.
Coprime Game and the Factor Graph
Definition
A factor graph is a graph with a consecutive set of integers for vertices, where i
and j are adjacent if they share a common
factor, other than 1.
Definition
A clique is a complete subgraph. It need not be maximal.
Coprime Game and the Factor Graph
Definition
A factor graph is a graph with a consecutive set of integers for vertices, where i
and j are adjacent if they share a common
factor, other than 1.
Definition
A clique is a complete subgraph. It need not be maximal.
The Posá problem shows that a factor graph with 2t
vertices has no clique of size t + 1.
Coprime Game and the Factor Graph
Definition
A factor graph is a graph with a consecutive set of integers for vertices, where i
and j are adjacent if they share a common
factor, other than 1.
Definition
A clique is a complete subgraph. It need not be maximal.
The Posá problem shows that a factor graph with 2t
vertices has no clique of size t + 1.
We saw that Bob won the coprime game.
Coprime Game and the Factor Graph
Definition
A factor graph is a graph with a consecutive set of integers for vertices, where i
and j are adjacent if they share a common
factor, other than 1.
Definition
A clique is a complete subgraph. It need not be maximal.
The Posá problem shows that a factor graph with 2t
vertices has no clique of size t + 1.
We saw that Bob won the coprime game.
Two Vertices Left Game
Game
Some graph G is given with an even number of vertices.
Two Vertices Left Game
Game
Some graph G is given with an even number of vertices.
Alice and Bob take turns deleting one vertex at a time,
with Alice moving first, until there are two vertices left.
Two Vertices Left Game
Game
Some graph G is given with an even number of vertices.
Alice and Bob take turns deleting one vertex at a time,
with Alice moving first, until there are two vertices left.
If they are adjacent, then Alice wins, and if they are
nonadjacent, then Bob wins.
Two Vertices Left Game
Game
Some graph G is given with an even number of vertices.
Alice and Bob take turns deleting one vertex at a time,
with Alice moving first, until there are two vertices left.
If they are adjacent, then Alice wins, and if they are
nonadjacent, then Bob wins.
Note: The Coprime Game is equivalent to the Two Vertices
Left Game played on an appropriate factor graph.
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Bob
Two Vertices Left Game
Example
Alice
Alice wins!
Bob
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
Alice removes a vertex u not on the clique.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Apply induction.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Apply induction.
Suppose G does not have a clique of size t + 1.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Apply induction.
Suppose G does not have a clique of size t + 1.
É
Alice removes a vertex u. G − u has 2t − 1 vertices.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Apply induction.
Suppose G does not have a clique of size t + 1.
É
É
Alice removes a vertex u. G − u has 2t − 1 vertices.
Bob removes a vertex v according to a lemma contained on
all cliques of size t.
Two Vertices Left Game Solution
Theorem
Alice wins the Two Vertices Left Game on a graph G with 2t
vertices if and only if G contains a clique on t + 1 vertices.
Proof.
(Modulo a lemma)
Suppose G has a clique of size t + 1.
É
É
É
Alice removes a vertex u not on the clique.
No matter what vertex v Bob removes, G − u − v will have a
clique of size t.
Apply induction.
Suppose G does not have a clique of size t + 1.
É
É
É
Alice removes a vertex u. G − u has 2t − 1 vertices.
Bob removes a vertex v according to a lemma contained on
all cliques of size t.
G − u − v will have no clique of size t. Apply induction.
The Helly-like Lemma
Lemma
Given a graph G on 2t − 1 vertices with no clique of size t + 1,
there exists a vertex v contained in every clique of size t.
Let k be the number of cliques of size t.
Helly-like Lemma “Proof"
Suppose k = 1.
Helly-like Lemma “Proof"
Suppose k = 1.
Helly-like Lemma “Proof"
Suppose k = 1.
Choose v to be any vertex on the clique.
Helly-like Lemma “Proof"
Suppose k = 2.
Helly-like Lemma “Proof"
Suppose k = 2.
The two cliques of size t cannot be disjoint, since G has
2t − 1 vertices.
Helly-like Lemma “Proof"
Suppose k = 2.
The two cliques of size t cannot be disjoint, since G has
2t − 1 vertices.
Let v be a vertex in the overlap.
Helly-like Lemma “Proof"
Suppose k = 3.
Helly-like Lemma “Proof"
Suppose k = 3.
Assume no vertex lies in all three cliques of size t.
Helly-like Lemma “Proof"
Suppose k = 3.
Assume no vertex lies in all three cliques of size t.
Let U be the vertices in pairwise intersections of the
cliques.
Helly-like Lemma “Proof"
Suppose k = 3.
Assume no vertex lies in all three cliques of size t.
Let U be the vertices in pairwise intersections of the
cliques.
U must be a clique.
Helly-like Lemma “Proof"
Suppose k = 3.
Assume no vertex lies in all three cliques of size t.
Let U be the vertices in pairwise intersections of the
cliques.
U must be a clique.
U has contains at least t + 1 vertices.
Helly-like Lemma “Proof"
Suppose k = 4.
Helly-like Lemma “Proof"
Suppose k = 4.
Helly-like Lemma “Proof"
Suppose k = 4.
Let U be one of the cliques
Helly-like Lemma “Proof"
Suppose k = 4.
Let U be one of the cliques
Form U0 from U by removing all vertices in no other
cliques, and add in the intersection of the other three
cliques.
Helly-like Lemma “Proof"
Suppose k = 4.
Let U be one of the cliques
Form U0 from U by removing all vertices in no other
cliques, and add in the intersection of the other three
cliques.
U0 forms a clique.
Helly-like Lemma “Proof"
Suppose k = 4.
Let U be one of the cliques
Form U0 from U by removing all vertices in no other
cliques, and add in the intersection of the other three
cliques.
U0 forms a clique.
By choosing the correct U to begin with, U0 will be larger
than U and hence be a clique of size at least t + 1.
Thanks!
Tyler Seacrest
University of Nebraska - Lincoln
[email protected]