Cycle cover property and CPP=SCC property are not equivalent

Abstract

Let G be an undirected graph. The Chinese postman problem (CPP) asks for a shortest postman tour in G, i.e., a closed walk using each edge at least once. The shortest cycle cover problem (SCC) asks for a family C of circuits of G such that each edge is in some circuit of C and the total length of all circuits in C is as small as possible. Clearly, an optimal solution of CPP cannot be greater than a solution of SCC. A graph G has the CPP = SCC property when the solutions to the two problems have the same value. Graph G is said to have the cycle cover property if for every Eulerian 1,2-weighting w : E(G) ↦ {1,2} there exists a family C of circuits of G such that every edge e is in precisely w e circuits of C . The cycle cover property implies the CPP = SCC property. We give a counterexample to a conjecture of Zhang (J. Graph Theory 14(5) (1990) 537; Ann. Discrete Math. 55 (1993) 183; Integer Flows and Cycle Covers of Graphs, Marcel Dekker, New York, 1997; Deans, Graph Structure Theory, AMS, Providence, RI, 1993, pp. 677–688) stating the equivalence of the cycle cover property and the CPP = SCC property for 3-connected graphs. This is also a counterexample to the stronger conjecture of Lai and Zhang, stating that every 3-connected graph with the CPP = SCC property has a nowhere-zero 4-flow. We actually obtain infinitely many cyclically 4-connected counterexamples to both conjectures.