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8Flows and circuit covers

Readers are referred to Appendix C for definitions and fundamental properties in the integer flow theory.

In this chapter, we present some results about circuit cover problems directly resulting from and related to integer flow theory.

8.1 Jaeger Theorems: 4-flow and 8-flow

Theorem 8.1.1 (Matthews [174]) Let r be a positive integer. A graph G admits a nowhere-zero 2r-flow if and only if G has an r-even subgraph cover.

Proof “⇒”: Induction on r. By Theorem C.2.3, let (D,f1) be a 2r-1-flow of G and (D, f2) be a 2-flow of G such that supp(f1) ∪ supp(f2) = E(G). By the inductive hypothesis, supp(f1) is covered by even subgraphs {C1,…, Cr–1}. Thus, {C1,…, Cr–1, Cr} is an r-even subgraph cover of G where Cr is the even subgraph ...

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