Cycle double cover conjecture

In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces.

It is an unsolved problem, posed by George Szekeres and Paul Seymour and known as the cycle double cover conjecture, whether every bridgeless graph has a cycle double cover. The conjecture can equivalently be formulated in terms of graph embeddings, and in that context is also known as the circular embedding conjecture.

The usual formulation of the cycle double cover conjecture asks whether every bridgeless undirected graph has a collection of cycles such that each edge of the graph is contained in exactly two of the cycles. The requirement that the graph be bridgeless is an obvious necessary condition for such a set of cycles to exist, because a bridge cannot belong to any cycle. A collection of cycles satisfying the condition of the cycle double cover conjecture is called a cycle double cover. Some graphs such as cycle graphs and bridgeless cactus graphs can only be covered by using the same cycle more than once, so this sort of duplication is allowed in a cycle double cover.

A snark is a special case of a bridgeless graph, having the additional properties that every vertex has exactly three incident edges (that is, the graph is cubic) and that it is not possible to partition the edges of the graph into three perfect matchings (that is, the graph has no 3-edge coloring, and by Vizing's theorem has chromatic index 4). It turns out that snarks form the only difficult case of the cycle double cover conjecture: if the conjecture is true for snarks, it is true for any graph.

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