Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a polygonal path composed of line segments with two adjacent directions from a given set $S$ of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply $NP$-hardness for any $S$ with $|S|\ge 4$ even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where $S$ contains precisely the horizontal and vertical directions, can be solved in polynomial time [GD'09]. Our results imply that this is incorrect unless $P=NP$. Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer and \v{S}tefankovi\v{c}. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Supereulerian Properties in Graphs and Hamiltonian Properties in Line Graphs

Following the trend initiated by Chvatal and Erdos, using the relation of independence number and connectivity as sufficient conditions for hamiltonicity of graphs, we characterize supereulerian graphs with small matching number, which implies a characterization of hamiltonian claw-free graph with small independence number.;We also investigate strongly spanning trailable graphs and their applications to hamiltonian connected line graphs characterizations for small strongly spanning trailable graphs and strongly spanning trailable graphs with short longest cycles are obtained. In particular, we have found a graph family F of reduced nonsupereulerian graphs such that for any graph G with kappa\u27(G) ≥ 2 and alpha\u27( G) ≤ 3, G is supereulerian if and only if the reduction of G is not in F..;We proved that any connected graph G with at most 12 vertices, at most one vertex of degree 2 and without vertices of degree 1 is either supereulerian or its reduction is one of six exceptional cases. This is applied to show that if a 3-edge-connected graph has the property that every pair of edges is joined by a longest path of length at most 8, then G is strongly spanning trailable if and only if G is not the wagner graph.;Using charge and discharge method, we prove that every 3-connected, essentially 10-connected line graph is hamiltonian connected. We also provide a unified treatment with short proofs for several former results by Fujisawa and Ota in [20], by Kaiser et al in [24], and by Pfender in [40]. New sufficient conditions for hamiltonian claw-free graphs are also obtained

On the Yang-Lee and Langer singularities in the O(n) loop model

We use the method of `coupling to 2d QG' to study the analytic properties of the universal specific free energy of the O(n) loop model in complex magnetic field. We compute the specific free energy on a dynamical lattice using the correspondence with a matrix model. The free energy has a pair of Yang-Lee edges on the high-temperature sheet and a Langer type branch cut on the low-temperature sheet. Our result confirms a conjecture by A. and Al. Zamolodchikov about the decay rate of the metastable vacuum in presence of Liouville gravity and gives strong evidence about the existence of a weakly metastable state and a Langer branch cut in the O(n) loop model on a flat lattice. Our results are compatible with the Fonseca-Zamolodchikov conjecture that the Yang-Lee edge appears as the nearest singularity under the Langer cut.Comment: 38 pages, 16 figure