Book contents
- Frontmatter
- Contents
- Foreword
- Foreword
- Preface
- 1 Circuit double cover
- 2 Faithful circuit cover
- 3 Circuit chain and Petersen minor
- 4 Small oddness
- 5 Spanning minor, Kotzig frames
- 6 Strong circuit double cover
- 7 Spanning trees, supereulerian graphs
- 8 Flows and circuit covers
- 9 Girth, embedding, small cover
- 10 Compatible circuit decompositions
- 11 Other circuit decompositions
- 12 Reductions of weights, coverages
- 13 Orientable cover
- 14 Shortest cycle covers
- 15 Beyond integer (1, 2)-weight
- 16 Petersen chain and Hamilton weights
- Appendix A Preliminary
- Appendix B Snarks, Petersen graph
- Appendix C Integer flow theory
- Appendix D Hints for exercises
- Glossary of terms and symbols
- References
- Author index
- Subject index
Foreword
Published online by Cambridge University Press: 05 May 2012
- Frontmatter
- Contents
- Foreword
- Foreword
- Preface
- 1 Circuit double cover
- 2 Faithful circuit cover
- 3 Circuit chain and Petersen minor
- 4 Small oddness
- 5 Spanning minor, Kotzig frames
- 6 Strong circuit double cover
- 7 Spanning trees, supereulerian graphs
- 8 Flows and circuit covers
- 9 Girth, embedding, small cover
- 10 Compatible circuit decompositions
- 11 Other circuit decompositions
- 12 Reductions of weights, coverages
- 13 Orientable cover
- 14 Shortest cycle covers
- 15 Beyond integer (1, 2)-weight
- 16 Petersen chain and Hamilton weights
- Appendix A Preliminary
- Appendix B Snarks, Petersen graph
- Appendix C Integer flow theory
- Appendix D Hints for exercises
- Glossary of terms and symbols
- References
- Author index
- Subject index
Summary
When I use the term multigraph decomposition, I mean a partition of the edge set. In particular, a cycle decomposition of a multigraph is a partition of the edge set into cycles, where I am using cycle to indicate a connected subgraph in which each vertex has valency 2.
There is a short list of cycle decomposition problems that I view as important problems. At the top of my list is the so-called cycle double cover conjecture which is the underlying motivation for this book. My reasons for ranking it at the top are discussed next.
If a conjecture has been largely ignored, then longevity essentially is irrelevant, but when a conjecture has been subjected to considerable research, then longevity plays a significant role in its importance. The cycle double cover conjecture has been with us for more than thirty years and has received considerable attention including three special workshops devoted to just this single conjecture. Thus, just in terms of longevity the cycle double cover conjecture acquires importance.
There is a deep, but not well understood, connection with the structure of graphs for if a graph X contains no Petersen minor, then a vast generalization of the cycle double conjecture is true. Trying to understand what is going on in this realm adds considerably to the allure of the cycle double cover conjecture.
Another strong attraction of the conjecture is the connections with other subareas of graph theory. These include topological graph theory, graph coloring, and flows in graphs.
- Type
- Chapter
- Information
- Circuit Double Cover of Graphs , pp. xiii - xivPublisher: Cambridge University PressPrint publication year: 2012