Circuit double cover of graphs (Q2884704)

The Circuit (Cycle) Double Cover Conjecture (CDC conjecture) states that every \(2\)-connected graph has a family \(\mathcal{F}\) of circuits such that every edge of the graph is covered by (contained in) precisely two members of \(\mathcal{F}\). (A circuit in this book is a connected \(2\)-regular graph.) This conjecture is one of the major open problems in graph theory. One reason for this is its close relationship with topological graph theory, integer flow theory, graph coloring and the structure of snarks.NEWLINENEWLINEThis book provides a thorough exploration of the general conjecture, the approaches that have been developed, and connections with different subareas. The author zooms in to focus on the rich body of results, methods and questions, directly and indirectly related to the CDC conjecture. The book is presented in a well-written form.NEWLINENEWLINEIn order to keep the main theme of the book as focused as possible, the author decided to concentrate for the most part on results and techniques in structural graph theory. Topological results are presented only if either they can be obtained by combinatorial methods or they are needed as background for a combinatorial approach.NEWLINENEWLINEWithout using flow theory, \(3\)-edge-coloring of cubic graphs becomes one of the major techniques in this book. Since flow theory does provide some powerful tools, it will be covered in later chapters.NEWLINENEWLINEMost of the basic lemmas and theorems of integer flow theory in this book are presented in an appendix. Moreover, a set of appendices collects some fundamental graph theoretical results which are useful in the study of circuit covers. The appendix also contains hints to the exercises and a glossary with basic and some special terminology.