Circuit Double Cover of GraphsThe famous Circuit Double Cover conjecture (and its numerous variants) is considered one of the major open problems in graph theory owing to its close relationship with topological graph theory, integer flow theory, graph coloring and the structure of snarks. It is easy to state: every 2-connected graph has a family of circuits covering every edge precisely twice. C.-Q. Zhang provides an up-to-date overview of the subject containing all of the techniques, methods and results developed to help solve the conjecture since the first publication of the subject in the 1940s. It is a useful survey for researchers already working on the problem and a fitting introduction for those just entering the field. The end-of-chapter exercises have been designed to challenge readers at every level and hints are provided in an appendix. |
Contents
Faithful circuit cover | 10 |
2 | 19 |
Open problems | 43 |
2 | 51 |
4 | 57 |
Exercises | 63 |
4 | 83 |
5 | 93 |
Orientable cover | 153 |
Shortest cycle covers | 163 |
Beyond integer 12weight | 189 |
Petersen chain and Hamilton weights | 199 |
Appendix A Preliminary | 243 |
Appendix B Snarks Petersen graph | 252 |
Integer flow theory | 273 |
Hints for exercises | 285 |
9 | 112 |
4 | 116 |
Compatible circuit decompositions | 117 |
Other circuit decompositions | 134 |
Glossary of terms and symbols | 322 |
References | 337 |
| 351 | |
Other editions - View all
Common terms and phrases
2-cell embedding 2-factor 3-even subgraph cover 5-even subgraph double admits a nowhere-zero bridgeless cubic graph bridgeless graph C₁ CDC conjecture circuit chain circuit decomposition circuit double cover circuit of G coloring component contradicts Corollary cover F cover of G cover problem Definition double cover conjecture edge of G edge-cut eulerian graph faithful circuit cover faithful cover Fano plane Figure Fleischner G contains graph G graph G admits graph obtained Graph Theory Hamilton circuit Hamilton path Heawood graph Hint for Exercise induced path integer flow Jaeger Kotzig graph L-graph Lemma Let F Let G Let H members of F Möbius ladders nowhere-zero 4-flow perfect matching permutation graph Petersen graph planar graph Proposition removable circuit semi-Kotzig snarks spanning even subgraph spanning subgraph spanning trees strong CDC subgraph double cover subgraph of G suppressed cubic graph Theorem Tutte uniquely 3-edge-colorable vertex vertices weighted graph