Directed cycle double covers: structure and generation of hexagon graphs

Jaeger's directed cycle double cover conjecture can be formulated as a problem of existence of special perfect matchings in a class of graphs that we call hexagon graphs. In this work, we explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig's augmentations.Comment: 20 page

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Jaeger’s Strong 3-Flow Conjecture for Graphs in Low Genus Surfaces

In 1972, Tutte posed the 3-Flow Conjecture: that all 4-edge-connected graphs have a nowhere zero 3-flow. This was extended by Jaeger et al. (1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo 3) between the inflow and outflow. He conjectured that all 5-edge-connected graphs with a valid prescription function have a nowhere zero 3-flow meeting that prescription. Kochol (2001) showed that replacing 4-edge-connected with 5-edge-connected would suffice to prove the 3-Flow Conjecture and Lovàsz et al. (2013) showed that both conjectures hold if the edge connectivity condition is relaxed to 6-edge-connected. Both problems are still open for 5-edge-connected graphs. The 3-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grötzsch's Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al. (2016) provided the first direct proof using flows of Jaeger's Strong 3-Flow Conjecture for planar graphs. We extend their result to graphs embedded in the projective plane. Lai (2007) showed that Jaeger's Strong 3-Flow Conjecture cannot be extended to 4-edge-connected graphs by constructing an infinite family of 4-edge-connected graphs that do not have a nowhere zero 3-flow meeting their prescribed net flow. We prove that graphs with arbitrarily many non-crossing 4-edge-cuts sufficiently far apart have a nowhere zero 3-flow, regardless of their prescription function. This is a step toward answering the question of which 4-edge-connected graphs have this property