Nash Williams Conjecture and the Dominating Cycle Conjecture

The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture

Nowhere-zero flows and structures in cubic graphs

Wir widerlegen zwei Vermutungen, die im Zusammenhang mit Kreisüberdeckungen von kubischen Graphen stehen. Die erste Vermutung, welche kubische Graphen mit dominierenden Kreisen betrifft, widerlegen wir durch Erweiterung eines Theorems von Gallai über induzierte eulersche Graphen und durch Konstruktion spezieller snarks. Die zweite Vermutung, welche frames betrifft, widerlegen wir durch Betrachtung der Frage nach der Existenz von speziellen spannenden Teilgraphen in 3-fach zusammenhängenden kubischen Graphen. Weiters übersetzen wir Probleme über Flüsse in kubischen Graphen in Knotenfärbungsprobleme von planaren Graphen und erhalten eine neue Charakterisierung von snarks. Schliesslich verbessern und erweitern wir Resultate über Knotenfärbungsprobleme in Quadrangulierungen. Zu Ende stellen wir neue Vermutungen auf, die im Zusammenhang mit Kreisüberdeckungen und Strukturen in kubischen Graphen stehen.We disprove two conjectures which are related to cycle double cover problems. The first conjecture concerns cubic graphs with dominating cycle. We disprove this conjecture by extending a result of Gallai about induced eulerian subgraphs and by constructing special snarks. The second conjecture concerns frames. We show that this conjecture is false by considering the problem whether every 3-connected cubic graph has a spanning subgraph with certain properties. Moreover, we transform flow-problems of cubic graphs into vertex coloring problems of plane graphs. We obtain thereby a new characterization of snarks. Furthermore, we improve and extend results about vertex coloring problems of quadrangulations. Finally we pose new problems and state conjectures which are related to cycle double covers and structures in cubic graphs

A note on 5-cycle double covers

The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which contains $C$, i.e. $C$ is a subgraph of one of the five 2-regular subgraphs of $S$. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of $G$