Hall's Theorem (marriage theorem) S1,S2,,S,n have a SDR→ VI C {1,2,...,m),Uier S
S1, S2,...,Sm have a SDR Hall’s Theorem (marriage theorem) ⇥I {1, 2,...,m}, ⇥ iI Si |I|
Hall's Theorem(graph theory form) A bipartite graph G(U,V,E)has a matching of U → lW(S)l≥ISl for all SCU matching:edge independent set MCE with no e1,e2CM share a vertex W(S)={vI3u∈S,uv∈E} V
Hall’s Theorem (graph theory form) A bipartite graph G(U,V,E) has a matching of U |N(S)| ≥ |S| for all S⊆U U V matching: edge independent set M⊆E with no e1,e2∈M share a vertex N(S) = {v | ∃u∈S, uv∈E}
Hall's Theorem(marriage theorem) I{1,2,,m,U∈rS≥|I S1,S2,...,Sm have a SDR critical family:S1,S2,...,S k<m k Us: =k i= Induction on m: m =1,trivial case.I:there is no critical family in S1,S2,...,S case.2:there is a critical family in S1,S2,...,Sm
S1, S2,...,Sm have a SDR Hall’s Theorem (marriage theorem) ⇥I {1, 2,...,m}, ⇥ iI Si |I|. Induction on m : m =1, trivial case.1: there is no critical family in S1, S2, ..., Sm case.2: there is a critical family in S1, S2, ..., Sm critical family: S1, S2,...,Sk ⇥ k i=1 Si = k k < m
Hall's Theorem(marriage theorem) VI C{1,2,...,m),Uier Si >S1,S2,...,Sm have a SDR case.I:there is no critical family in S1,S2,...,Sm I∈{1,2，，m}that|<m,Jier S>lIl take an arbitrary xESm as representative of S remove Sm and x Si'=Slx}i=1,2,.,m-1 I≤{1,2，，m-1},UerS≥I due to I.H.S1,....Sm-1 have a SDR 1,...,m-1} 1,....m-1 and form a SDR for S1,S2,...,Sm
case.1: there is no critical family in S1, S2, ..., Sm take an arbitrary x∈Sm as representative of Sm remove Sm and x Si ’ = Si\{x} i = 1, 2, ..., m-1 ⌅I ⇥ {1, 2,...,m 1}, ⇥ i⇥I S i ⇤ |I| ⇥I {1, 2,...,m} that |I| < m, ⇥ iI Si > |I| due to I.H. S1, S2,...,Sm have a SDR Hall’s Theorem (marriage theorem) ⇥I {1, 2,...,m}, ⇥ iI Si |I|. S⇥ 1,...,S⇥ m1 have a SDR {x1,...,xm1} x1,...,xm1 and x form a SDR for S1, S2, ..., Sm
Hall's Theorem(marriage theorem) VI C{1,2,...,m),Uier Si. S1:S2,...,Sm have a SDR case.2: there is a critical family in S1,S2,...,S say|Sm-k+1U·USml=k k<m due to I.H.Sm-k+1,..,Sm have a SDR X=x1,..xx Si'-SX i=1,2,.,m-k I≤{1,2，..，m-k}, Um-k+1S:UUieI Sik+ Uier S≥II|
S1, S2,...,Sm have a SDR Hall’s Theorem (marriage theorem) ⇥I {1, 2,...,m}, ⇥ iI Si |I|. case.2: there is a critical family in S1, S2, ..., Sm say k < m due to I.H. Sm-k+1,..., Sm have a SDR X={x1, ..., xk} |Smk+1 ⇥ ··· ⇥ Sm| = k ⇤I ⇥ {1, 2,...,m k}, ⇥m i=mk+1 Si ⇥ i⇥I Si k + |I| |I| ⇥ i⇥I S i Si ’ = Si\X i = 1, 2, ..., m-k