![](/files/happy5.png)
Cycle Double Covers Containing Predefined 2-Regular Subgraphs
Conjecture Let
be a
-connected cubic graph and let
be a
-regular subgraph such that
is connected. Then
has a cycle double cover which contains
(i.e all cycles of
).
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ G-E(S) $](/files/tex/f65af23afc4481cc9a13687d7a6d12108bda7714.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
Used definitions in the above conjecture: a "cycle" is a connected 2-regular subgraph, a "cycle double cover" of a graph is a set of cycles of
such that every edge of
is contained in precisely two cycles of the set. This conjecture has been motivated by Theorem 3, respectively, Theorem 4 in www.arxiv.org/abs/1711.10614. A weaker conjecture (Conjecture 14) has been stated in "Snarks with special spanning trees" (see www.arxiv.org/abs/1706.05595).