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Matchings and factors in graphs Part 5

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Page 1: Matchings and factors in graphs

Matchings and factors in graphs

Part 5

Page 2: Matchings and factors in graphs

f -factorsFor a m/graph G and a function f : V (G)→ N, an f -factor of Gis a spanning subgraph F of G s.t. dF (v) = f (v) ∀v ∈ V (G).

Given a m/graph G and a function f : V (G)→ N, an f -blowupH(G) of G is obtained from G as follows:

(a) Each vertex v is replaced by the complete bipartite graphwith parts A(v) and B(v) where |A(v)| = d(v) and|B(v)| = d(v)− f (v);

(b) for every edge e ∈ E(G) connecting x ∈ V (G) withy ∈ V (G), e connects in H(G) a vertex in A(x) with a vertex inA(y) so that no two distinct edges of G have a common end inH(G).

Page 3: Matchings and factors in graphs

f -factorsFor a m/graph G and a function f : V (G)→ N, an f -factor of Gis a spanning subgraph F of G s.t. dF (v) = f (v) ∀v ∈ V (G).

Given a m/graph G and a function f : V (G)→ N, an f -blowupH(G) of G is obtained from G as follows:

(a) Each vertex v is replaced by the complete bipartite graphwith parts A(v) and B(v) where |A(v)| = d(v) and|B(v)| = d(v)− f (v);

(b) for every edge e ∈ E(G) connecting x ∈ V (G) withy ∈ V (G), e connects in H(G) a vertex in A(x) with a vertex inA(y) so that no two distinct edges of G have a common end inH(G).

Page 4: Matchings and factors in graphs

f -factorsFor a m/graph G and a function f : V (G)→ N, an f -factor of Gis a spanning subgraph F of G s.t. dF (v) = f (v) ∀v ∈ V (G).

Given a m/graph G and a function f : V (G)→ N, an f -blowupH(G) of G is obtained from G as follows:

(a) Each vertex v is replaced by the complete bipartite graphwith parts A(v) and B(v) where |A(v)| = d(v) and|B(v)| = d(v)− f (v);

(b) for every edge e ∈ E(G) connecting x ∈ V (G) withy ∈ V (G), e connects in H(G) a vertex in A(x) with a vertex inA(y) so that no two distinct edges of G have a common end inH(G).

Page 5: Matchings and factors in graphs

Theorem 1.19 (Tutte): A m/graph G has an f -factor iff anf -blowup H(G) of G has a p.m.

Page 6: Matchings and factors in graphs

Theorem 1.19 (Tutte): A m/graph G has an f -factor iff anf -blowup H(G) of G has a p.m.

Page 7: Matchings and factors in graphs

Some notation: for S ⊆ V (G), f (S) =∑

v∈S f (v).

Theorem 1.20 (Belck, 1950): A multigraph G has a k -factor ifand only if

q(S,T )− dG−S(T ) ≤ k(|S| − |T |) (1)

for all disjoint S,T ⊂ V (G), where q(S,T ) is the number ofcomponents Q of G−S − T such that |EG(V (Q),T )|+ k |V (Q)|is odd.

Why (1) is necessary? What about k = 1?

More notation for f -factors: for disjoint S,T ⊆ V (G), letR = V (G)− S − T .A component Q of G[R] is f -odd (or simply odd) if

f (Q) + |EG(Q,T )| is odd.

Let q(S,T ) be the number of odd components of G[R].

Page 8: Matchings and factors in graphs

Some notation: for S ⊆ V (G), f (S) =∑

v∈S f (v).

Theorem 1.20 (Belck, 1950): A multigraph G has a k -factor ifand only if

q(S,T )− dG−S(T ) ≤ k(|S| − |T |) (1)

for all disjoint S,T ⊂ V (G), where q(S,T ) is the number ofcomponents Q of G−S − T such that |EG(V (Q),T )|+ k |V (Q)|is odd.

Why (1) is necessary? What about k = 1?

More notation for f -factors: for disjoint S,T ⊆ V (G), letR = V (G)− S − T .A component Q of G[R] is f -odd (or simply odd) if

f (Q) + |EG(Q,T )| is odd.

Let q(S,T ) be the number of odd components of G[R].

Page 9: Matchings and factors in graphs

Some notation: for S ⊆ V (G), f (S) =∑

v∈S f (v).

Theorem 1.20 (Belck, 1950): A multigraph G has a k -factor ifand only if

q(S,T )− dG−S(T ) ≤ k(|S| − |T |) (1)

for all disjoint S,T ⊂ V (G), where q(S,T ) is the number ofcomponents Q of G−S − T such that |EG(V (Q),T )|+ k |V (Q)|is odd.

Why (1) is necessary? What about k = 1?

More notation for f -factors: for disjoint S,T ⊆ V (G), letR = V (G)− S − T .A component Q of G[R] is f -odd (or simply odd) if

f (Q) + |EG(Q,T )| is odd.

Let q(S,T ) be the number of odd components of G[R].

Page 10: Matchings and factors in graphs

Theorem 1.21 (Tutte, 1952, 1954): A multigraph G has anf -factor if and only if

q(S,T )− dG−S(T ) ≤ f (S)− f (T ) (2)

for all disjoint S,T ⊂ V (G).

Proof: The ”only if” part is simpler. So we prove the ”if” part.

Consider the f -blowup H of G. If H has a p.m., then we aredone by Theorem 1.19. Otherwise, consider a minimum Tutteset C in H.

A parasite is a vertex in C that has neighbors in at most twoodd components of H − C.

We first prove 4 claims (proofs in the book).

Page 11: Matchings and factors in graphs

Theorem 1.21 (Tutte, 1952, 1954): A multigraph G has anf -factor if and only if

q(S,T )− dG−S(T ) ≤ f (S)− f (T ) (2)

for all disjoint S,T ⊂ V (G).

Proof: The ”only if” part is simpler. So we prove the ”if” part.

Consider the f -blowup H of G. If H has a p.m., then we aredone by Theorem 1.19. Otherwise, consider a minimum Tutteset C in H.

A parasite is a vertex in C that has neighbors in at most twoodd components of H − C.

We first prove 4 claims (proofs in the book).

Page 12: Matchings and factors in graphs

Theorem 1.21 (Tutte, 1952, 1954): A multigraph G has anf -factor if and only if

q(S,T )− dG−S(T ) ≤ f (S)− f (T ) (2)

for all disjoint S,T ⊂ V (G).

Proof: The ”only if” part is simpler. So we prove the ”if” part.

Consider the f -blowup H of G. If H has a p.m., then we aredone by Theorem 1.19. Otherwise, consider a minimum Tutteset C in H.

A parasite is a vertex in C that has neighbors in at most twoodd components of H − C.

We first prove 4 claims (proofs in the book).

Page 13: Matchings and factors in graphs

Claim 1: C has no parasites.

Claim 2: ∀v ∈ V (G), either B(v) ⊆ C or B(v) ∩ C = ∅.

Claim 3: If B(v) ⊆ C or B(v) = ∅, then A(v) ∩ C = ∅.

Claim 4: If B(v) 6= ∅ and B(v) ∩ C = ∅, then eitherA(v) ∩ C = ∅ or A(v) ⊆ C.

Let T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅}(then A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (then B(v) ∩ C = ∅),R = V (G)− S − T .

Page 14: Matchings and factors in graphs

Claim 1: C has no parasites.

Claim 2: ∀v ∈ V (G), either B(v) ⊆ C or B(v) ∩ C = ∅.

Claim 3: If B(v) ⊆ C or B(v) = ∅, then A(v) ∩ C = ∅.

Claim 4: If B(v) 6= ∅ and B(v) ∩ C = ∅, then eitherA(v) ∩ C = ∅ or A(v) ⊆ C.

Let T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅}(then A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (then B(v) ∩ C = ∅),R = V (G)− S − T .

Page 15: Matchings and factors in graphs

Claim 1: C has no parasites.

Claim 2: ∀v ∈ V (G), either B(v) ⊆ C or B(v) ∩ C = ∅.

Claim 3: If B(v) ⊆ C or B(v) = ∅, then A(v) ∩ C = ∅.

Claim 4: If B(v) 6= ∅ and B(v) ∩ C = ∅, then eitherA(v) ∩ C = ∅ or A(v) ⊆ C.

Let T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅}(then A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (then B(v) ∩ C = ∅),R = V (G)− S − T .

Page 16: Matchings and factors in graphs

Claim 1: C has no parasites.

Claim 2: ∀v ∈ V (G), either B(v) ⊆ C or B(v) ∩ C = ∅.

Claim 3: If B(v) ⊆ C or B(v) = ∅, then A(v) ∩ C = ∅.

Claim 4: If B(v) 6= ∅ and B(v) ∩ C = ∅, then eitherA(v) ∩ C = ∅ or A(v) ⊆ C.

Let T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅}(then A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (then B(v) ∩ C = ∅),R = V (G)− S − T .

Page 17: Matchings and factors in graphs

Claim 1: C has no parasites.

Claim 2: ∀v ∈ V (G), either B(v) ⊆ C or B(v) ∩ C = ∅.

Claim 3: If B(v) ⊆ C or B(v) = ∅, then A(v) ∩ C = ∅.

Claim 4: If B(v) 6= ∅ and B(v) ∩ C = ∅, then eitherA(v) ∩ C = ∅ or A(v) ⊆ C.

Let T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅}(then A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (then B(v) ∩ C = ∅),R = V (G)− S − T .

Page 18: Matchings and factors in graphs

T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅} (and A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (and B(v) ∩ C = ∅).

Types of components of H − C:

Type 1: y ∈ B(v) for v ∈ S.

Type 2: {y , z} with yz ∈ E(H), y ∈ A(v),z ∈ A(u), uv ∈ E(G[T ]).

Type 3: y ∈ A(v), for all uv ∈ E(G([S,T ]).

Type 4: One for each component of G − S − T .

Page 19: Matchings and factors in graphs

T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅} (and A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (and B(v) ∩ C = ∅).

Types of components of H − C:

Type 1: y ∈ B(v) for v ∈ S.

Type 2: {y , z} with yz ∈ E(H), y ∈ A(v),z ∈ A(u), uv ∈ E(G[T ]).

Type 3: y ∈ A(v), for all uv ∈ E(G([S,T ]).

Type 4: One for each component of G − S − T .

Page 20: Matchings and factors in graphs

T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅} (and A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (and B(v) ∩ C = ∅).

Types of components: # of odd such comp-s:

Type 1: y ∈ B(v) for v ∈ S.∑

v∈S(d(v)− f (v)) = d(S)− f (S)

Type 2: {y , z} with yz ∈ E(H), y ∈ A(v),z ∈ A(u), uv ∈ E(G[T ]. 0

Type 3: y ∈ A(v), for all uv ∈ E(G([S,T ]). |EG(S,T )|

Type 4: One for each component of G − S − T . q(S,T )

Page 21: Matchings and factors in graphs

T = {v ∈ V (G) : B(v) ⊆ C or B(v) = ∅} (and A(v) ∩ C = ∅),S = {v ∈ V (G) : A(v) ⊆ C} (and B(v) ∩ C = ∅).

Types of components: # of odd such comp-s:

Type 1: y ∈ B(v) for v ∈ S.∑

v∈S(d(v)− f (v)) = d(S)− f (S)

Type 2: {y , z} with yz ∈ E(H), y ∈ A(v),z ∈ A(u), uv ∈ E(G[T ]. 0

Type 3: y ∈ A(v), for all uv ∈ E(G([S,T ]). |EG(S,T )|

Type 4: One for each component of G − S − T . q(S,T )

Page 22: Matchings and factors in graphs

By definition,

|C| =∑v∈S

d(v)+∑w∈T

(d(w)− f (w)) = dG(S)+dG(T )− f (T ). (3)

By (3) and the choice of C,

0 < o(H − C)− |C|

=(d(S)− f (S) + |E(S,T )|+ q(S,T )

)−(d(S) + d(T )− f (T )

).

This means

f (T ) > f (S)− |E(S,T )| − q(S,T ) + d(T )

= f (S)− q(S,T ) + dG−S(T ).

Page 23: Matchings and factors in graphs

By definition,

|C| =∑v∈S

d(v)+∑w∈T

(d(w)− f (w)) = dG(S)+dG(T )− f (T ). (3)

By (3) and the choice of C,

0 < o(H − C)− |C|

=(d(S)− f (S) + |E(S,T )|+ q(S,T )

)−(d(S) + d(T )− f (T )

).

This means

f (T ) > f (S)− |E(S,T )| − q(S,T ) + d(T )

= f (S)− q(S,T ) + dG−S(T ).

Page 24: Matchings and factors in graphs

By definition,

|C| =∑v∈S

d(v)+∑w∈T

(d(w)− f (w)) = dG(S)+dG(T )− f (T ). (3)

By (3) and the choice of C,

0 < o(H − C)− |C|

=(d(S)− f (S) + |E(S,T )|+ q(S,T )

)−(d(S) + d(T )− f (T )

).

This means

f (T ) > f (S)− |E(S,T )| − q(S,T ) + d(T )

= f (S)− q(S,T ) + dG−S(T ).

Page 25: Matchings and factors in graphs

Another Parity LemmaLemma 1.22: For all f : V (G)→ Z+ and disjoint S,T ⊂ V (G),Ff (S,T ) = f (S) + dG−S(T )− q(S,T )− f (T ) + f (V (G)) is even.

Proof: Let P(Y ) denote the parity of Y . Then

Pf (F (S,T )) = P(f (R))+P(dG−S(T ))+∑Q∈R

P(f (Q)+ |E(Q,T )|)

= P(dG−S(T )) + P(|E(R,T )|) = P(2|E(G[T ])|+ 2|E(R,T )|).

Page 26: Matchings and factors in graphs

Another Parity LemmaLemma 1.22: For all f : V (G)→ Z+ and disjoint S,T ⊂ V (G),Ff (S,T ) = f (S) + dG−S(T )− q(S,T )− f (T ) + f (V (G)) is even.

Proof: Let P(Y ) denote the parity of Y . Then

Pf (F (S,T )) = P(f (R))+P(dG−S(T ))+∑Q∈R

P(f (Q)+ |E(Q,T )|)

= P(dG−S(T )) + P(|E(R,T )|) = P(2|E(G[T ])|+ 2|E(R,T )|).

Page 27: Matchings and factors in graphs

Theorem 1.23 (Erdos and Gallai): A non-increasing sequenced = (d1, . . . ,dn) of nonnegative integers is graphic iffd1 + . . .+ dn is even and

k∑i=1

di ≤ k(k − 1) +n∑

j=k+1

min{k ,dj} ∀k ∈ [n]. (4)

Proof: The ”only if” part is easy. Suppose d is notgraphic although d1 + . . .+ dn is even and (4) holds.

Then Kn has no f -factor where f (vi) = di for i ∈ [n]. ByTheorem 1.21, ∃ disjoint S,T s.t.

f (T ) > f (S) + dKn−S(T )− q(S,T ).

So, by Lemma 1.22,

f (T ) ≥ 2 + f (S) + dKn−S(T )− q(S,T ). (5)

Page 28: Matchings and factors in graphs

Theorem 1.23 (Erdos and Gallai): A non-increasing sequenced = (d1, . . . ,dn) of nonnegative integers is graphic iffd1 + . . .+ dn is even and

k∑i=1

di ≤ k(k − 1) +n∑

j=k+1

min{k ,dj} ∀k ∈ [n]. (4)

Proof: The ”only if” part is easy. Suppose d is notgraphic although d1 + . . .+ dn is even and (4) holds.

Then Kn has no f -factor where f (vi) = di for i ∈ [n]. ByTheorem 1.21, ∃ disjoint S,T s.t.

f (T ) > f (S) + dKn−S(T )− q(S,T ).

So, by Lemma 1.22,

f (T ) ≥ 2 + f (S) + dKn−S(T )− q(S,T ). (5)

Page 29: Matchings and factors in graphs

Theorem 1.23 (Erdos and Gallai): A non-increasing sequenced = (d1, . . . ,dn) of nonnegative integers is graphic iffd1 + . . .+ dn is even and

k∑i=1

di ≤ k(k − 1) +n∑

j=k+1

min{k ,dj} ∀k ∈ [n]. (4)

Proof: The ”only if” part is easy. Suppose d is notgraphic although d1 + . . .+ dn is even and (4) holds.

Then Kn has no f -factor where f (vi) = di for i ∈ [n]. ByTheorem 1.21, ∃ disjoint S,T s.t.

f (T ) > f (S) + dKn−S(T )− q(S,T ).

So, by Lemma 1.22,

f (T ) ≥ 2 + f (S) + dKn−S(T )− q(S,T ). (5)

Page 30: Matchings and factors in graphs

Theorem 1.23 (Erdos and Gallai): A non-increasing sequenced = (d1, . . . ,dn) of nonnegative integers is graphic iffd1 + . . .+ dn is even and

k∑i=1

di ≤ k(k − 1) +n∑

j=k+1

min{k ,dj} ∀k ∈ [n]. (4)

Proof: The ”only if” part is easy. Suppose d is notgraphic although d1 + . . .+ dn is even and (4) holds.

Then Kn has no f -factor where f (vi) = di for i ∈ [n]. ByTheorem 1.21, ∃ disjoint S,T s.t.

f (T ) > f (S) + dKn−S(T )− q(S,T ).

So, by Lemma 1.22,

f (T ) ≥ 2 + f (S) + dKn−S(T )− q(S,T ). (5)

Page 31: Matchings and factors in graphs

Since G = Kn, (a) q(S,T ) ≤ 1, and(b) dKn−S(T ) = |T |(n − 1− |S|).

Hence (5) yields

f (T ) ≥ f (S) + |T |(n − 1− |S|) + 1. (6)

Denoting k = |T | and s = |S|, (5) yields

k∑i=1

di ≥n∑

j=n−s+1

dj + k(n − 1− s) + 1

=n∑

j=n−s+1

dj +n−s∑

j=k+1

k + (k − 1)k + 1,

contradicting (4).

Page 32: Matchings and factors in graphs

Since G = Kn, (a) q(S,T ) ≤ 1, and(b) dKn−S(T ) = |T |(n − 1− |S|).

Hence (5) yields

f (T ) ≥ f (S) + |T |(n − 1− |S|) + 1. (6)

Denoting k = |T | and s = |S|, (5) yields

k∑i=1

di ≥n∑

j=n−s+1

dj + k(n − 1− s) + 1

=n∑

j=n−s+1

dj +n−s∑

j=k+1

k + (k − 1)k + 1,

contradicting (4).