Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

Perfect Matching and Circuit Cover of Graphs

The research of my dissertation is motivated by the Circuit Double Cover Conjecture due to Szekeres and independently Seymour, that every bridgeless graph G has a family of circuits which covers every edge of G twice. By Fleischner\u27s Splitting Lemma, it suffices to verify the circuit double cover conjecture for bridgeless cubic graphs.;It is well known that every edge-3-colorable cubic graph has a circuit double cover. The structures of edge-3-colorable cubic graphs have strong connections with the circuit double cover conjecture. In chapter two, we consider the structure properties of a special class of edge-3-colorable cubic graphs, which has an edge contained by a unique perfect matching. In chapter three, we prove that if a cubic graph G containing a subdivision of a special class of edge-3-colorable cubic graphs, semi-Kotzig graphs, then G has a circuit double cover.;Circuit extension is an approach posted by Seymour to attack the circuit double cover conjecture. But Fleischer and Kochol found counterexamples to this approach. In chapter four, we post a modified approach, called circuit extension sequence. If a cubic graph G has a circuit extension sequence, then G has a circuit double cover. We verify that all Fleischner\u27s examples and Kochol\u27s examples have a circuit extension sequence, and hence not counterexamples to our approach. Further, we prove that a circuit C of a bridgeless cubic G is extendable if the attachments of all odd Tutte-bridges appear on C consequently.;In the last chapter, we consider the properties of minimum counterexamples to the strong circuit double cover. Applying these properties, we show that if a cubic graph G has a long circuit with at least | V(G)| - 7 vertices, then G has a circuit double cover