Cycle Cover Property and CPP=SCC Property are not Equivalent

Rizzi, Romeo (2002) Cycle Cover Property and CPP=SCC Property are not Equivalent. UNSPECIFIED.

Download (240Kb) | Preview


    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 can not 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 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.

    Item Type: Departmental Technical Report
    Department or Research center: Information Engineering and Computer Science
    Subjects: Q Science > QA Mathematics > QA075 Electronic computers. Computer science
    Uncontrolled Keywords: cycle cover, faithful cover, Petersen graph, 4-flow, counterexample
    Additional Information: Published: Discrete Mathematics 259 (2002) 337--342.
    Report Number: DIT-02-103
    Repository staff approval on: 29 Jan 2003

    Actions (login required)

    View Item