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Published online by Cambridge University Press: 05 May 2012
The Circuit (Cycle) Double Cover Conjecture (CDC conjecture) is easy to state: For every 2-connected graph, there is a family ℱ of circuits such that every edge of the graph is covered by precisely two members of ℱ. As an example, if a 2-connected graph is properly embedded on a surface (without crossing edges) in such a way that all faces are bounded by circuits, then the collection of the boundary circuits will “double cover” the graph.
The CDC conjecture (and its numerous variants) is considered by most graph theorists to be one of the major open problems in the field. One reason for this is its close relationship with topological graph theory, integer flow theory, graph coloring and the structure of snarks.
This long standing open problem has been discussed independently in various publications, such as G. Szekeres (1973 [219]), A. Itai and M. Rodeh (1978 [119]), and P. D. Seymour (1979 [205]). According to Professor W. T. Tutte, “the conjecture is one that was well established in mathematical conversation long before anyone thought of publishing it.” Some early investigations related to the conjecture can be traced back to publications by Tutte in the later 1940s.
Some material about circuit covers was presented in the book Integer Flows and Cycle Covers of Graphs (1997 [259]) by the author as an application of flow theory. There are several reasons why the author decided to write a follow-up book mainly on this subject.
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