Game f -matching in Graphs Douglas B. West Zhejiang Normal - - PowerPoint PPT Presentation

Game f-matching in Graphs

Douglas B. West

Zhejiang Normal University and University of Illinois at Urbana-Champaign west@math.uiuc.edu

slides available on DBW preprint page

Joint work with Jennifer I. Wise plus prior work with James Carraher, Daniel W. Cranston, William B. Kinnersley, Benjamin Reiniger, and Suil O

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex .

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex . (f-factor is subgraph with dH() = f() for all .)

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex . (f-factor is subgraph with dH() = f() for all .) We want a large f-matching but can’t pick all the edges.

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex . (f-factor is subgraph with dH() = f() for all .) We want a large f-matching but can’t pick all the edges. Game: T wo players, Max and Min, alternately add edges until chosen subgraph is a maximal f-matching. Max wants it to be large; Min wants it to be small.

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex . (f-factor is subgraph with dH() = f() for all .) We want a large f-matching but can’t pick all the edges. Game: T wo players, Max and Min, alternately add edges until chosen subgraph is a maximal f-matching. Max wants it to be large; Min wants it to be small. Final size is at most mf (G), the max size of f-matching, and at least mf(G), the min size of maximal f-matching.

The f-Matching Game

An f-matching in G is a subgraph H with dH() ≤ f() for each vertex . (f-factor is subgraph with dH() = f() for all .) We want a large f-matching but can’t pick all the edges. Game: T wo players, Max and Min, alternately add edges until chosen subgraph is a maximal f-matching. Max wants it to be large; Min wants it to be small. Final size is at most mf (G), the max size of f-matching, and at least mf(G), the min size of maximal f-matching.

• Def. νg(G) = outcome under best play if Max starts.

ˆ νg(G) = outcome under best play if Min starts.

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F.

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F. F-saturation game - Max/Min play continues until the chosen subgraph is F-saturated.

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F. F-saturation game - Max/Min play continues until the chosen subgraph is F-saturated. When f() = r for all , the f-matching game is the same as the K1,r+1-saturation game.

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F. F-saturation game - Max/Min play continues until the chosen subgraph is F-saturated. When f() = r for all , the f-matching game is the same as the K1,r+1-saturation game. Under optimal play, outcome is stg(G; F) for the Max-start game, stg(G; F) for the Min-start game.

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F. F-saturation game - Max/Min play continues until the chosen subgraph is F-saturated. When f() = r for all , the f-matching game is the same as the K1,r+1-saturation game. Under optimal play, outcome is stg(G; F) for the Max-start game, stg(G; F) for the Min-start game. Always st(G; F) ≤ stg(G; F) ≤ ex(G; F); similarly for stg(G; F).

Saturation Games

F-saturated subgraph of G - a maximal subgraph among those not “having” F. F-saturation game - Max/Min play continues until the chosen subgraph is F-saturated. When f() = r for all , the f-matching game is the same as the K1,r+1-saturation game. Under optimal play, outcome is stg(G; F) for the Max-start game, stg(G; F) for the Min-start game. Always st(G; F) ≤ stg(G; F) ≤ ex(G; F); similarly for stg(G; F). Does the choice of starting player really matter?

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.
• t edges if Max starts.

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.
• t edges if Max starts.

2 edges if Min starts.

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.
• t edges if Max starts.

2 edges if Min starts.

• Ex. (P2 + P3)-saturation in tP3.

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.
• t edges if Max starts.

2 edges if Min starts.

• Ex. (P2 + P3)-saturation in tP3.
• 2 edges if Max starts.

Max-start vs. Min-start in Saturation Games

• Ex. (P2 + P2)-saturation in near-K1,t.
• t edges if Max starts.

2 edges if Min starts.

• Ex. (P2 + P3)-saturation in tP3.
• 2 edges if Max starts.

t edges if Min starts.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all .

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

• Thm. All positive pairs (, j) except (1, 2) occur.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

• Thm. All positive pairs (, j) except (1, 2) occur.
• Thm. ν1(G) ≥ 2

3m1(G), which is sharp.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

• Thm. All positive pairs (, j) except (1, 2) occur.
• Thm. ν1(G) ≥ 2

3m1(G), which is sharp.

• Thm. ν1(G) ≤ min

3

2m1(G), m1(G)

• , which is sharp.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

• Thm. All positive pairs (, j) except (1, 2) occur.
• Thm. ν1(G) ≥ 2

3m1(G), which is sharp.

• Thm. ν1(G) ≤ min

3

2m1(G), m1(G)

• , which is sharp.
• Thm. ν1(G) ≥ 3

4m1(G) when G is a forest (sharp).

Also ˆ ν1(G) ≤ ν1(G) for forests.

The 1-Matching Game

Write νk(G) and ˆ νk(G) when f() = k for all . Prior results (Cranston–Kinnersley–O–West [2012]):

• Thm. |ν1(G) − ˆ

ν1(G)| ≤ 1 for every graph G.

• Thm. If H is an induced subgraph of G, then

ν1(H) ≤ ν1(G) and ˆ ν1(H) ≤ ˆ ν1(G).

• Thm. All positive pairs (, j) except (1, 2) occur.
• Thm. ν1(G) ≥ 2

3m1(G), which is sharp.

• Thm. ν1(G) ≤ min

3

2m1(G), m1(G)

• , which is sharp.
• Thm. ν1(G) ≥ 3

4m1(G) when G is a forest (sharp).

Also ˆ ν1(G) ≤ ν1(G) for forests. Which extend to f-matching or at least 2-matching?

The Easy Lower Bound Generalizes

• Thm. νf (G) ≥ 2

3mf(G) for every graph G.

The Easy Lower Bound Generalizes

• Thm. νf (G) ≥ 2

3mf(G) for every graph G.

• Pf. Consider a round of play.

The Easy Lower Bound Generalizes

• Thm. νf (G) ≥ 2

3mf(G) for every graph G.

• Pf. Consider a round of play.

Max plays an edge  of a maximum f-matching M. Let G′ = G −  and f ′ = {, }-reduction of f. Note mf ′(G′) ≥ mf (G) − 1.

The Easy Lower Bound Generalizes

• Thm. νf (G) ≥ 2

3mf(G) for every graph G.

• Pf. Consider a round of play.

Max plays an edge  of a maximum f-matching M. Let G′ = G −  and f ′ = {, }-reduction of f. Note mf ′(G′) ≥ mf (G) − 1. Min plays some edge y. Let G′′ = G′ − y and f ′′ = {, y}-reduction of f ′. Deleting edges from M′ at  and y leaves f ′′-matching. Thus mf ′′(G′′) ≥ mf ′(G′) − 2.

• −1

−1

The Easy Lower Bound Generalizes

• Thm. νf (G) ≥ 2

3mf(G) for every graph G.

• Pf. Consider a round of play.

Max plays an edge  of a maximum f-matching M. Let G′ = G −  and f ′ = {, }-reduction of f. Note mf ′(G′) ≥ mf (G) − 1. Min plays some edge y. Let G′′ = G′ − y and f ′′ = {, y}-reduction of f ′. Deleting edges from M′ at  and y leaves f ′′-matching. Thus mf ′′(G′′) ≥ mf ′(G′) − 2.

• −1

−1

• ∴ Each round plays 2 edges and reduces max size of

achievable subgraph by at most 3.

Sharpness

• Ex. Let G = Kn

Kn and f() = k for all  ∈ V(G). Note mk(G) = kn, mk(G) = 1

2kn,

νk(G) = 2

3kn.

S T

• k

k k k k k

Sharpness

• Ex. Let G = Kn

Kn and f() = k for all  ∈ V(G). Note mk(G) = kn, mk(G) = 1

2kn,

νk(G) = 2

3kn.

S T

• k

k k k k k Min can reduce capacity in T by 2 with each move.

Sharpness

• Ex. Let G = Kn

Kn and f() = k for all  ∈ V(G). Note mk(G) = kn, mk(G) = 1

2kn,

νk(G) = 2

3kn.

S T

• k

k k k k k Min can reduce capacity in T by 2 with each move.

• Always mf(G) ≥ 1

2mf(G).

Sharpness

• Ex. Let G = Kn

Kn and f() = k for all  ∈ V(G). Note mk(G) = kn, mk(G) = 1

2kn,

νk(G) = 2

3kn.

S T

• k

k k k k k Min can reduce capacity in T by 2 with each move.

• Always mf(G) ≥ 1

2mf(G).

Uses that an f-matching M is a maximum f-matching if and only if G has no M-augmenting trail.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2. Max plays some edge, reducing M-degree by ≥ 1.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2. Max plays some edge, reducing M-degree by ≥ 1. Reducing M-degree by 2 kills one or two edges of M.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2. Max plays some edge, reducing M-degree by ≥ 1. Reducing M-degree by 2 kills one or two edges of M. Min plays in M for r rounds, s killing 3 edges: 2r+s≥|M|. These 2r moves reduce M-degree by at least 3r + s.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2. Max plays some edge, reducing M-degree by ≥ 1. Reducing M-degree by 2 kills one or two edges of M. Min plays in M for r rounds, s killing 3 edges: 2r+s≥|M|. These 2r moves reduce M-degree by at least 3r + s. Remaining M-degree (and moves) are ≤ 2|M| − 3r − s.

The Easy Upper Bound Generalizes

• Thm. νf (G) ≤ 3

2mf(G) for every graph G.

• Pf. Let M be a smallest maximal f-matching.

Vertices with f() > dM() are nonadjacent or joined by an edge of M, by maximality. Min plays in M when possible, reducing M-degree by 2. Max plays some edge, reducing M-degree by ≥ 1. Reducing M-degree by 2 kills one or two edges of M. Min plays in M for r rounds, s killing 3 edges: 2r+s≥|M|. These 2r moves reduce M-degree by at least 3r + s. Remaining M-degree (and moves) are ≤ 2|M| − 3r − s. Hence νf (G) ≤ 2|M| − r − s ≤ 3

2|M|,

since 2r + s ≥ |M| implies r + s ≥ |M|/2.

Sharpness

• Ex. Let G consist of t copies of P4 with

edge-multiplicity k (with kt even) and f() = k for all . Here mf (G) = kt and νf (G) = 3kt/2.

Sharpness

• Ex. Let G consist of t copies of P4 with

edge-multiplicity k (with kt even) and f() = k for all . Here mf (G) = kt and νf (G) = 3kt/2.

• Ex. Let G consist of Kk+1 with k pendant edges at each

vertex, and f() = k for all ; νf (G) = 3

2

k+1

2

.

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move. Key: If f ≡ 1, then νf ′(G − ) = ν1(G − {, }).

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move. Key: If f ≡ 1, then νf ′(G − ) = ν1(G − {, }).

• Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V(G),

(1) |ν1(G) − ˆ ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ).

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move. Key: If f ≡ 1, then νf ′(G − ) = ν1(G − {, }).

• Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V(G),

(1) |ν1(G) − ˆ ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ).

• Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G.

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move. Key: If f ≡ 1, then νf ′(G − ) = ν1(G − {, }).

• Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V(G),

(1) |ν1(G) − ˆ ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ).

• Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G.

Let  and y be optimal starts for Max and Min on G. ν1(G) = 1 + ˆ ν1(G −  − ) ≤ 1 + ˆ ν1(G − ) ≤ 1 + ˆ ν1(G). ˆ ν1(G) = 1 + ν1(G −  − y) ≤ 1 + ν1(G − y) ≤ 1 + ν1(G).

Always |ν1(G) − ˆ ν1(G)| ≤ 1

• Def. S-reduction of f reduces capacity by 1 for  ∈ S.
• Prop. If  ∈ E(G) and f ′ is {, }-reduction of f, then

νf (G) ≥ 1 + ˆ νf ′(G − ) and ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move. Key: If f ≡ 1, then νf ′(G − ) = ν1(G − {, }).

• Thm. (Cranston–Kinnersley–O–West [2012]) For  ∈ V(G),

(1) |ν1(G) − ˆ ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ).

• Pf. Step 1: (2) holds for G & smaller ⇒ (1) holds for G.

Let  and y be optimal starts for Max and Min on G. ν1(G) = 1 + ˆ ν1(G −  − ) ≤ 1 + ˆ ν1(G − ) ≤ 1 + ˆ ν1(G). ˆ ν1(G) = 1 + ν1(G −  − y) ≤ 1 + ν1(G − y) ≤ 1 + ν1(G). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G.

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ).

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G.

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G. Let H = G − . Let y be optimal start for Max on H. ν1(G) ≥ 1 + ˆ ν1(G −  − y) ≥ 1 + ˆ ν1(H −  − y) = ν1(H).

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G. Let H = G − . Let y be optimal start for Max on H. ν1(G) ≥ 1 + ˆ ν1(G −  − y) ≥ 1 + ˆ ν1(H −  − y) = ν1(H). Let y be optimal start for Min on G. If  / ∈ {, y}, then ˆ ν1(G) = 1 + ν1(G −  − y) ≥ 1 + ν1(H −  − y) ≥ ˆ ν1(H).

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G. Let H = G − . Let y be optimal start for Max on H. ν1(G) ≥ 1 + ˆ ν1(G −  − y) ≥ 1 + ˆ ν1(H −  − y) = ν1(H). Let y be optimal start for Min on G. If  / ∈ {, y}, then ˆ ν1(G) = 1 + ν1(G −  − y) ≥ 1 + ν1(H −  − y) ≥ ˆ ν1(H). If  =  and z ∈ NH(y), then using first move yz in H, ˆ ν1(G) = 1 + ν1(G −  − y) ≥ 1 + ν1(G −  − y − z) ≥ ˆ ν1(H).

Completion of proof

• Prop. νf (G) ≥ 1 + ˆ

νf ′(G − ) & ˆ νf (G) ≤ 1 + νf ′(G − ). Equality ⇔  is optimal first move.

• Thm. (1) |ν1(G) − ˆ

ν1(G)| ≤ 1, and (2) ν1(G) ≥ ν1(G − ) and ˆ ν1(G) ≥ ˆ ν1(G − ). Step 2: (1) & (2) hold for smaller ⇒ (2) holds for G. Let H = G − . Let y be optimal start for Max on H. ν1(G) ≥ 1 + ˆ ν1(G −  − y) ≥ 1 + ˆ ν1(H −  − y) = ν1(H). Let y be optimal start for Min on G. If  / ∈ {, y}, then ˆ ν1(G) = 1 + ν1(G −  − y) ≥ 1 + ν1(H −  − y) ≥ ˆ ν1(H). If  =  and z ∈ NH(y), then using first move yz in H, ˆ ν1(G) = 1 + ν1(G −  − y) ≥ 1 + ν1(G −  − y − z) ≥ ˆ ν1(H). If  =  and dH(y) = 0, then ˆ ν1(G) = 1+ν1(G−−y) = 1+ν1(H−y) = 1+ν1(H) ≥ ˆ ν1(H).

Open Problem

• Conj. (1) |νf (G) − ˆ

νf (G)| ≤ 1 for general G and f.

Open Problem

• Conj. (1) |νf (G) − ˆ

νf (G)| ≤ 1 for general G and f. Possible added monotonicity statements for induction:

Open Problem

• Conj. (1) |νf (G) − ˆ

νf (G)| ≤ 1 for general G and f. Possible added monotonicity statements for induction:

• (2) If f() ≥ 1, and h is the {}-reduction of f,

then νf (G) ≥ νh(G) and ˆ νf (G) ≥ ˆ νh(G).

Open Problem

• Conj. (1) |νf (G) − ˆ

νf (G)| ≤ 1 for general G and f. Possible added monotonicity statements for induction:

• (2) If f() ≥ 1, and h is the {}-reduction of f,

then νf (G) ≥ νh(G) and ˆ νf (G) ≥ ˆ νh(G).

• (2) If f(), f() ≥ 1, and f ′ is the {, }-reduction of f,

then νf (G) ≥ νf ′(G − ) and ˆ νf (G) ≥ ˆ νf ′(G − ).

Open Problem

• Conj. (1) |νf (G) − ˆ

νf (G)| ≤ 1 for general G and f. Possible added monotonicity statements for induction:

• (2) If f() ≥ 1, and h is the {}-reduction of f,

then νf (G) ≥ νh(G) and ˆ νf (G) ≥ ˆ νh(G).

• (2) If f(), f() ≥ 1, and f ′ is the {, }-reduction of f,

then νf (G) ≥ νf ′(G − ) and ˆ νf (G) ≥ ˆ νf ′(G − ). Parts of the argument for 1-matching generalize, but it seems harder to use these to prove (1).

An Easy Directed Version

Idea: Impose capacity f() only on the outdegree of .

An Easy Directed Version

Idea: Impose capacity f() only on the outdegree of . Given undirected G, players Max and Min alternately select and orient an edge for the subgraph H so that always d+

H() ≤ f() at each  ∈ V(G).

An Easy Directed Version

Idea: Impose capacity f() only on the outdegree of . Given undirected G, players Max and Min alternately select and orient an edge for the subgraph H so that always d+

H() ≤ f() at each  ∈ V(G).

They aim to maximize and minimize the final |E(H)|, respectively.

An Easy Directed Version

Idea: Impose capacity f() only on the outdegree of . Given undirected G, players Max and Min alternately select and orient an edge for the subgraph H so that always d+

H() ≤ f() at each  ∈ V(G).

They aim to maximize and minimize the final |E(H)|, respectively. Let µf(G) and ˆ µf (G) denote the number of edges selected under optimal play in the Max-start and Min-start games, respectively.

An Easy Directed Version

Idea: Impose capacity f() only on the outdegree of . Given undirected G, players Max and Min alternately select and orient an edge for the subgraph H so that always d+

H() ≤ f() at each  ∈ V(G).

They aim to maximize and minimize the final |E(H)|, respectively. Let µf(G) and ˆ µf (G) denote the number of edges selected under optimal play in the Max-start and Min-start games, respectively.

• Thm. For every graph G and capacity function f on G,

|µf(G) − ˆ µf(G)| ≤ 1.

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G).

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G). Let Y consist of f() copies of  for all  ∈ V(G).

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G). Let Y consist of f() copies of  for all  ∈ V(G). Make  ∈ X adjacent in G′ to all copies in Y of  and .

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G). Let Y consist of f() copies of  for all  ∈ V(G). Make  ∈ X adjacent in G′ to all copies in Y of  and . Since |ν1(G′) − ˆ ν1(G′)| ≤ 1, it suffices to show µf (G) = ν1(G′) and ˆ µf(G) = ˆ ν1(G′).

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G). Let Y consist of f() copies of  for all  ∈ V(G). Make  ∈ X adjacent in G′ to all copies in Y of  and . Since |ν1(G′) − ˆ ν1(G′)| ≤ 1, it suffices to show µf (G) = ν1(G′) and ˆ µf(G) = ˆ ν1(G′). Selecting e oriented away from  in the directed f-matching game on G corresponds to picking edge e′ in the 1-matching game on G′ for some copy ′ of .

Transformation Argument

• Thm. |µf (G) − ˆ

µf(G)| ≤ 1 for all G and capacity f.

• Pf. Build auxiliary (X, Y)-bigraph G′.

Let X = E(G). Let Y consist of f() copies of  for all  ∈ V(G). Make  ∈ X adjacent in G′ to all copies in Y of  and . Since |ν1(G′) − ˆ ν1(G′)| ≤ 1, it suffices to show µf (G) = ν1(G′) and ˆ µf(G) = ˆ ν1(G′). Selecting e oriented away from  in the directed f-matching game on G corresponds to picking edge e′ in the 1-matching game on G′ for some copy ′ of . Each e ∈ E(G) = X is selected at most once. Each  ∈ V(G) is made tail (matched) ≤ f() times.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

• Obs. The 2-matching game on Kn is near-fair, since

maximal 2-matchings have n − 1 or n edges.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

• Obs. The 2-matching game on Kn is near-fair, since

maximal 2-matchings have n − 1 or n edges. Thm.

(Carraher–Kinnersley–Reiniger–West [2013+]) For

n ≥ 5 (and n = 7), always ν2(Kn) = ˆ ν2(Kn), with Player 1 “winning” for even n and Player 2 “winning” for odd n.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

• Obs. The 2-matching game on Kn is near-fair, since

maximal 2-matchings have n − 1 or n edges. Thm.

(Carraher–Kinnersley–Reiniger–West [2013+]) For

n ≥ 5 (and n = 7), always ν2(Kn) = ˆ ν2(Kn), with Player 1 “winning” for even n and Player 2 “winning” for odd n.

• Thm. (Wise–West) If G =
• Kn with all n having the

same parity (and not in {1, 2, 3, 4, 7}), then G is near-fair. Also, the values are

• (n − 1) and

1 +

• (n − 1), with Player 2 winning unless G consists of

an odd number of even-order components.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

• Obs. The 2-matching game on Kn is near-fair, since

maximal 2-matchings have n − 1 or n edges. Thm.

(Carraher–Kinnersley–Reiniger–West [2013+]) For

n ≥ 5 (and n = 7), always ν2(Kn) = ˆ ν2(Kn), with Player 1 “winning” for even n and Player 2 “winning” for odd n.

• Thm. (Wise–West) If G =
• Kn with all n having the

same parity (and not in {1, 2, 3, 4, 7}), then G is near-fair. Also, the values are

• (n − 1) and

1 +

• (n − 1), with Player 2 winning unless G consists of

an odd number of even-order components. Uses edge-transitivity of Kn.

Another Question

• Def. G with capacity f is near-fair if |νf (G)− ˆ

νf (G)| ≤ 1.

• Ques. When G and H with capacities are near-fair,

under what conditions must G + H be near-fair?

• Obs. The 2-matching game on Kn is near-fair, since

maximal 2-matchings have n − 1 or n edges. Thm.

(Carraher–Kinnersley–Reiniger–West [2013+]) For

n ≥ 5 (and n = 7), always ν2(Kn) = ˆ ν2(Kn), with Player 1 “winning” for even n and Player 2 “winning” for odd n.

• Thm. (Wise–West) If G =
• Kn with all n having the

same parity (and not in {1, 2, 3, 4, 7}), then G is near-fair. Also, the values are

• (n − 1) and

1 +

• (n − 1), with Player 2 winning unless G consists of

an odd number of even-order components. Uses edge-transitivity of Kn. Unions of components with different parities are hard to handle.

Back to Saturation Games

The k-matching game on G is the same as the K1,k+1-saturation game on G.

Back to Saturation Games

The k-matching game on G is the same as the K1,k+1-saturation game on G. Some results on saturation games: (G; F) st(G; F) s = stg(G; F) ex(G; F) (Kn, K3) n − 1 Ω(n lg n) ≤ s ≤ n2/5? n2/4 (Kn; P4) n/2 ≈ 4n/5 n or n − 1 (Km,n; P4) n − 2    n n even m + n−1

2

mn odd m else = stg(G; F) (Kn,n; C4) n − 1 s > Ω(n13/12) n3/2 + O(n4/3)

Füredi–Reimer–Seress [1991] for lower bound on st(Kn; K3). Carraher–Kinnersley–Reiniger–West [2013+] for others.

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

• Lem. Let G be C4-saturated in Kn,n with parts X and Y.

If ∃ S ⊆ V(G) and c, d such that |S ∩ X|, |S ∩ Y| ≥ cn and |NG() ∩ S| ≤ dn for all , then |E(G)| ≥ n13/12, where  = min{ 1

2( c2 2d2 )2/3, c2 2d}.

X Y

• S ∩ X

S ∩ Y ≥ cn ≥ cn ≤ dn

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

• Lem. Let G be C4-saturated in Kn,n with parts X and Y.

If ∃ S ⊆ V(G) and c, d such that |S ∩ X|, |S ∩ Y| ≥ cn and |NG() ∩ S| ≤ dn for all , then |E(G)| ≥ n13/12, where  = min{ 1

2( c2 2d2 )2/3, c2 2d}.

X Y

• S ∩ X

S ∩ Y y  ≥ cn ≥ cn C4-sat ⇒ ≥ c2n2 − cdn3/2 such S-paths.

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

• Lem. Let G be C4-saturated in Kn,n with parts X and Y.

If ∃ S ⊆ V(G) and c, d such that |S ∩ X|, |S ∩ Y| ≥ cn and |NG() ∩ S| ≤ dn for all , then |E(G)| ≥ n13/12, where  = min{ 1

2( c2 2d2 )2/3, c2 2d}.

X Y

• S ∩ X

S ∩ Y y  ≥ cn ≥ cn C4-sat ⇒ ≥ c2n2 − cdn3/2 such S-paths. In half, central edge has endpt of degree < n5/12.

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

• Lem. Let G be C4-saturated in Kn,n with parts X and Y.

If ∃ S ⊆ V(G) and c, d such that |S ∩ X|, |S ∩ Y| ≥ cn and |NG() ∩ S| ≤ dn for all , then |E(G)| ≥ n13/12, where  = min{ 1

2( c2 2d2 )2/3, c2 2d}.

X Y

• S ∩ X

S ∩ Y y  ≥ cn ≥ cn C4-sat ⇒ ≥ c2n2 − cdn3/2 such S-paths. In half, central edge has endpt of degree < n5/12. Each such is central edge for at most dn11/12 S-paths.

Sketch of Lower Bound

Idea: Max constructs a subgraph of Kn,n whose C4-saturated supergraphs have Ω(n13/12) edges.

• Lem. Let G be C4-saturated in Kn,n with parts X and Y.

If ∃ S ⊆ V(G) and c, d such that |S ∩ X|, |S ∩ Y| ≥ cn and |NG() ∩ S| ≤ dn for all , then |E(G)| ≥ n13/12, where  = min{ 1

2( c2 2d2 )2/3, c2 2d}.

X Y

• S ∩ X

S ∩ Y y  ≥ cn ≥ cn C4-sat ⇒ ≥ c2n2 − cdn3/2 such S-paths. In half, central edge has endpt of degree < n5/12. Each such is central edge for at most dn11/12 S-paths. ∴ at least c2

2dn13/12 such edges.

Max Strategy

• Thm. stg(Kn,n; C4) ≥

1 10.4n13/12.

Max Strategy

• Thm. stg(Kn,n; C4) ≥

1 10.4n13/12.

• Pf. In first 2n/3 moves, Max gives degree k to k

specified vertices in each part, where k =

• n/3
• − 1,

by joining them to isolated vertices on the other side. X Y

• • • •
• • • •
• • • •
• • • •

Max Strategy

• Thm. stg(Kn,n; C4) ≥

1 10.4n13/12.

• Pf. In first 2n/3 moves, Max gives degree k to k

specified vertices in each part, where k =

• n/3
• − 1,

by joining them to isolated vertices on the other side. X Y

• • • •
• • • •
• • • •
• • • •

Since G has no 4-cycle, each vertex has at most one leaf neighbor in each star.

Max Strategy

• Thm. stg(Kn,n; C4) ≥

1 10.4n13/12.

• Pf. In first 2n/3 moves, Max gives degree k to k

specified vertices in each part, where k =

• n/3
• − 1,

by joining them to isolated vertices on the other side. X Y

• • • •
• • • •
• • • •
• • • •

Since G has no 4-cycle, each vertex has at most one leaf neighbor in each star. ∴ With c ≈ 1/3 and d =

• 1/3, the conditions of the

lemma hold.

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

• Thm. For 3-regular connected n-vertex graphs with

perfect matchings, n/3 ≤ min ν1(G) ≤ 3n/7.

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

• Thm. For 3-regular connected n-vertex graphs with

perfect matchings, n/3 ≤ min ν1(G) ≤ 3n/7.

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

• Thm. For 3-regular connected n-vertex graphs with

perfect matchings, n/3 ≤ min ν1(G) ≤ 3n/7.

• Thm. For 3-regular connected n-vertex graphs,

n/3 ≤ min ν1(G) ≤ 7n/18.

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

• Thm. For 3-regular connected n-vertex graphs with

perfect matchings, n/3 ≤ min ν1(G) ≤ 3n/7.

• Thm. For 3-regular connected n-vertex graphs,

n/3 ≤ min ν1(G) ≤ 7n/18. This page has 7s.

One More Open Problem

• Ques. For 3-regular connected n-vertex graphs with

perfect matchings, how small can ν1(G) be?

• Thm. For 3-regular connected n-vertex graphs with

perfect matchings, n/3 ≤ min ν1(G) ≤ 3n/7.

• Thm. For 3-regular connected n-vertex graphs,

n/3 ≤ min ν1(G) ≤ 7n/18. This page has 7s.

Happy Birthday, Bjarne!