Stronger Conjectures and Related Problems
A stronger version of the circular embedding conjecture that has also been considered is the conjecture that every biconnected graph has a circular embedding on an orientable manifold. In terms of the cycle double cover conjecture, this is equivalent to the conjecture that there exists a cycle double cover, and an orientation for each of the cycles in the cover, such that for every edge e the two cycles that cover e are oriented in opposite directions through e.
Alternatively, strengthenings of the conjecture that involve colorings of the cycles in the cover have also been considered. The strongest of these is a conjecture that every bridgeless graph has a circular embedding on an orientable manifold in which the faces can be 5-colored. If true, this would imply a conjecture of W. T. Tutte that every bridgeless graph has a nowhere-zero 5-flow.
A stronger type of embedding than a circular embedding is a polyhedral embedding, an embedding of a graph on a surface in such a way that every face is a simple cycle and every two faces that intersect do so in either a single vertex or a single edge. (In the case of a cubic graph, this can be simplified to a requirement that every two faces that intersect do so in a single edge.) Thus, in view of the reduction of the cycle double cover conjecture to snarks, it is of interest to investigate polyhedral embeddings of snarks. Unable to find such embeddings, Branko Grünbaum conjectured that they do not exist, but Kochol (2009a, 2009b) disproved Grünbaum's conjecture by finding a snark with a polyhedral embedding.
Read more about this topic: Cycle Double Cover
Famous quotes containing the words stronger, conjectures, related and/or problems:
“O, my offense is rank, it smells to heaven;
It hath the primal eldest curse upon ‘t,
A brother’s murder. Pray can I not,
Though inclination be as sharp as will;
My stronger guilt defeats my strong intent,
And like a man to double business bound
I stand in pause where I shall first begin,
And both neglect. What if this cursed hand
Were thicker than itself with brother’s blood,
Is there not rain enough in the sweet heavens
To wash it white as snow?”
—William Shakespeare (1564–1616)
“Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“One does not realize the historical sensation as a re-experiencing, but as an understanding that is closely related to the understanding of music, or rather of the world by means of music.”
—Johan Huizinga (1872–1945)
“Our [adult] children have an adult’s right to make their own choices and have the responsibility of living with the consequences. If we make their problems ours, they avoid that responsibility, and we are faced with problems we can’t and shouldn’t solve.”
—Jane Adams (20th century)