Planar Hypohamiltonian Graphs on 40 Vertices
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer‐aided generation of certain families of planar graphs with girth 4 and a fixed number of 4‐faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices.
New results on hypohamiltonian and almost hypohamiltonian graphs
Consider a non-hamiltonian graph $G$. $G$ is hypohamiltonian if for every vertex $v$ in $G$, the graph $G - v$ is hamiltonian. $G$ is almost hypohamiltonian if there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$. McKay asked in [J. Graph Theory, doi: 10.1002/jgt.22043] whether infinitely many planar cubic hypohamiltonian graphs of girth 5 exist. We settle this question affirmatively. 
The second author asked in [J. Graph Theory 79 (2015) 63--81] to find all orders for which almost hypohamiltonian graphs exist---we solve this problem as well. To this end, we present a specialised algorithm which generates complete sets of almost hypohamiltonian graphs for various orders. Furthermore, we show that the smallest cubic almost hypohamiltonian graphs have order 26. We also provide a lower bound for the order of the smallest planar almost hypohamiltonian graph and improve the upper bound for the order of the smallest planar almost hypohamiltonian graph containing a cubic vertex. Finally, we determine the smallest planar almost hypohamiltonian graphs of girth 5, both in the general and cubic case.
Synthesis of quasi-stationary inhomogeneous impedance body of arbitrary shape
Formulas were obtained for the reflection factors of a plane wave from the inhomogeneous impedance plane, which reflects the incident wave in the given direction. The problem of synthesis of impedance inhomogeneous rotation body was solved for an arbitrary shaped generatrix for the given scattering diagram for co- and cross polarizations. Impedance distributions were obtained in explicit form. Restrictions on the class of implementable scat- terplots were formulated.