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CSORE4010 HOMEWORK 6 due on March 23 at the start of the class 1. Let G be a digraph and for each edge e let (e) 0 be an integer, so that for every...

I got problem 4, but none of the rest.
CSORE4010— HOMEWORK 6 due on March 23 at the start of the class 1. Let G be a digraph and for each edge e let φ ( e ) 0 be an integer, so that for every vertex v , s e δ - ( v ) φ ( e ) = s e δ + ( v ) φ ( e ) Show there is a list C 1 ,...,C n of directed cycles (possibly with repeti- tion) so that for every edge e of G , |{ i : 1 i n, e E ( C i ) }| = φ ( e ) . 2. Let s,t be distinct vertices of a digraph G , and let c : E ( G ) Z + . Let φ be a c -admissible s - t Fow of value k and suppose there is a c -admissible Fow with value > k . Does it follow that there is a c - admissible Fow ψ of value > k so that φ ( e ) ψ ( e ) for every edge e ? 3. Let s,t be vertices of a digraph G , and let φ : E ( G ) IR + be an s - t Fow. Show that there is an s - t Fow ψ : E ( G ) ZZ + so that (a) its value is at least that of φ , and (b) | ψ ( e ) - φ ( e ) | < 1 for every edge e of G . 4. Let T be a tree. Show that T has a perfect matching if and only if for every vertex v , exactly one of the components of T \ v has an odd number of vertices. [Hint for “if”: use induction on | V ( T ) | , ±nd a leaf with a neighbour of degree 2, and delete them both.] 1
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