Parity Subgraph, Shortest Cycle Cover, and Postman Tour

Abstract

Let $G = ( V,E )$ be a simple graph such that the number of odd vertices of G is $| V_0 |$ and the minimum odd degree is $\delta _0 $. This paper proves that the number of edges in a smallest parity subgraph of G is at most $| V | - {\text{Min}} \{ \delta _{0} , | V | - | V_0 | /2 \}$. Consequently, some results about the shortest cycle cover problem due to Itai and Rodeh, Fan, Zhang, Raspaud, Zhao are generalized. If G is a 2-edge-connected simple graph such that either G admits a nowhere-zero 4-flow or G contains no subdivision of the Petersen graph, then the total length of a shortest cycle cover of G is at most $| E | + | V | - {\text{Min}} \{ \delta _{0} , | V | - | V_0 |/2 \}$.