No abstract

A graph G with n vertices is Hamiltonian if it admits an embedded cycle containing all vertices of G. In any Hamiltonian graph, each vertex is the starting point of a Hamiltonian path. In this paper we explore the converse. We show that for 2 < n < 9, if G admits Hamiltonian paths starting at every vertex then G is Hamiltonian. We also show that this is not true for n > 9. We then investigate the number of pairs of vertices in a non-Hamiltonian graph G which can be connected by Hamiltonian paths. In particular we construct a family of non-Hamiltonian graphs with approximately 4/5 of the pairs of vertices connected by Hamiltonian paths.

Recent research into analog computing has introduced new notions of computing real numbers. Huang, Klinge, Lathrop, Li, and Lutz defined a notion of computing real numbers in real-time with chemical reaction networks (CRNs), introducing the classes RLCRN (the class of all Lyapunov CRN-computable real numbers) and RRTCRN (the class of all real-time CRN-computable numbers). In their paper, they show the inclusion of the real algebraic numbers ALG Ď RLCRN Ď RRTCRN and that ALG Ř RRTCRN but leave open where the inclusion is proper. In this paper, we resolve this open problem and show ALG " RLCRN Ř RRTCRN. However, their definition of real-time computation is fragile in the sense that it is sensitive to perturbations in initial conditions. To resolve this flaw, we further require a CRN to withstand these perturbations. In doing so, we arrive at a discrete model of memory. This approach has several benefits. First, a bounded CRN may compute values approximately in finite time. Second, a CRN can tolerate small perturbations of its species' concentrations. Third, taking a measurement of a CRN's state only requires precision proportional to the exactness of these approximations. Lastly, if a CRN requires only finite memory, this model and Turing machines are equivalent under real-time simulations.

A 3-graph is a pair H = (V, E) of sets, where elements of V are called points or vertices and E contains some 3-element subsets of V , called edges. A 3-graph is called linear if any two distinct edges intersect in at most one vertex.There is a recent interest in extremal properties of 3-graphs containing no crown, three pairwise disjoint edges and a fourth edge which intersects all of them. We show that every linear 3-graph with minimum degree 4 contains a crown. This is not true if 4 is replaced by 3.