On hypohamiltonian snarks and a theorem of Fiorini

In 2003, Cavicchioli et al. corrected an omission in the statement and proof of Fiorini's theorem from 1983 on hypohamiltonian snarks. However, their version of this theorem contains an unattainable condition for certain cases. We discuss and extend the results of Fiorini and Cavicchioli et al. and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen had shown that hypohamiltonian snarks exist for some orders n >= 10 and each even n >= 92. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exist. This also strengthens a result of Macajova and Skoviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices

Martingale approach to stochastic differential games of control and stopping

We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.Comment: Published in at http://dx.doi.org/10.1214/07-AOP367 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Polyhedra with few 3-cuts are hamiltonian

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each $k \ge 6$ non-hamiltonian polyhedra with $k$ 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl

Every graph occurs as an induced subgraph of some hypohamiltonian graph

We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K_3, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph

On almost hypohamiltonian graphs

A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717

Goal Programming as a Decision Model for Performance-Based Budgeting

AbstractThe spreadsheet applications are becoming a widely used tool for processing quantitative data and developing analytical solutions. The ability to build decision support systems (DSS) based on these spreadsheet solutions can facilitate knowledge management and increase information utilization within an organization. The paper describes the use of a spreadsheet solution to partially implement the performance-based budgeting (PB) principles for a public body. The solution builds on the goal programming optimization method and is exemplified for some real data

Every point is critical

We show that, for any compact Alexandrov surface S (without boundary) and any point y in S, there exists a point x in S for which y is a critical point. Moreover, we prove that uniqueness characterizes the surfaces homeomorphic to the sphere among smooth orientable surfaces. © 2013