11 - Other circuit decompositions

Summary

Restricted circuit decompositions

By looking at the incorrect proof presented in Section 10.1, one may suggest some different restrictions so that some circuit decompositions of eulerian graphs may imply the CDC conjecture.

Example 1 Let H be the even graph obtained from a bridgeless cubic graph G by adding a path of length 2 between every pair {x, y} if xy ∈ E(G). If H has a circuit decomposition ℱ such that every member of ℱ is of even length, then ℱ corresponds to a circuit double cover of the original graph G.

Example 2 Let H be the even graph described in the previous example. If H has a circuit decomposition ℱ′ such that no member of ℱ is of length 3, then ℱ′ corresponds to a circuit double cover of the original graph G.

Those two examples motivated some related studies of special circuit decompositions of eulerian graphs.

Theorem 11.1.1 (Seymour [206]) If H is a planar even graph such that every block contains an even number of edges, then H has a circuit decomposition consisting of even circuits.

Theorem 11.1.2 (Zhang [255]) If H is a K₅-minor-free even graph such that every block contains an even number of edges, then H has a circuit decomposition consisting of even circuits.

Theorem 11.1.2 is an application of Theorem 10.3.3. A proof of this theorem different from the original paper [255] can be found in [259] which uses a removable circuit theorem in [87] and, therefore, it is an application of both Theorems 10.3.3 and 12.1.2.