On the number of subgroups of given index in the modular group

A new recurrence for the number of subgroups of given index in the modular group is derived. The proof requires the derivation of a recurrence for a sequencea _nb_n from recurrences for thea _n andb _n. We show that this is always possible if thea _n andb _n satisfy polynomial recurrences. We also include a short proof of a result ofW. W. Stothers on the parity of the number of subgroups of given index in the modular group.     

On crossing numbers of hypercubes and cube connected cycles

We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs.The research of both authors was supported by Alexander von Humboldt Foundation, Bonn, Germany.     

The Same Upper Bound for Both: The 2‐page and the Rectilinear Crossing Numbers of the n‐Cube

We present two main results: a 2‐page and a rectilinear drawing of the n‐dimensional cube . Both drawings have the same number of crossings, even though they are given by different constructions. The first improves the current best general 2‐page drawing, while the second is the first nontrivial rectilinear drawing of .     

Edge-transitive products

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a strong product is edge-transitive if and only if all factors are complete graphs. In addition, a connected, infinite non-trivial Cartesian product graph G is edge-transitive if and only if it is vertex-transitive and if G is a finite weak Cartesian power of a connected, edge- and vertex-transitive graph H, or if G is the weak Cartesian power of a connected, bipartite, edge-transitive graph H that is not vertex-transitive.     

Regioselective syntheses, structural characterization, and electron ionization mass spectrometric behavior of regioisomeric 2,3-disubstituted 2-imino-1,3-thiazolidin-4-ones

The regioselective syntheses of 3-alkyl(aryl)-2-(anthracen-9'-ylimino)-1,3-thiazolidin-4-ones (2) and 2-alkyl(aryl)imino-3-(anthracen-9'-yl)-1,3-thiazolidin-4-ones (3) from N-(anthracen-9-yl)-N'-alkyl(aryl)thioureas were accomplished effectively using methyl bromoacetate and bromoacetyl bromide, respectively. Detailed structural characteristics were confirmed mainly by NMR techniques. The mass spectrometric behavior of the resulting sets of compounds of known structures was shown to be characteristic for each set. Some interesting fragmentation pathways involving the transfer and rearrangements of various moieties were also revealed, as well as regioisomerization for particular substituent-specific fragmentations.     

Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

The biplanar crossing number cr₂(G) of a graph G is min{cr(G₁) + cr(G₂)}, where cr is the planar crossing number. We show that cr₂(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) ‐ 2 ≤ Kcr₂(G)^0.4057 log₂n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr₂(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log² n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m²/k²) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m³/n²k²), provided that m = Ψ(n²). © 1996 John Wiley & Sons, Inc.     

4‐(9,10‐Dihydroacridin‐9‐ylidene)thiosemicarbazide and its five‐membered thiazole and six‐membered thiazine derivatives

Two methyl derivatives, five‐membered methyl 2‐{2‐[2‐(9,10‐dihydroacridin‐9‐ylidene)‐1‐methylhydrazinyl]‐4‐oxo‐4,5‐dihydro‐1,3‐thiazol‐5‐ylidene}acetate, C₂₀H₁₆N₄O₃S, (I), and six‐membered 2‐[2‐(9,10‐dihydroacridin‐9‐ylidene)‐1‐methylhydrazinyl]‐4H‐1,3‐thiazin‐4‐one, C₁₈H₁₄N₄OS, (II), were prepared by the reaction of the N‐methyl derivative of 4‐(9,10‐dihydroacridin‐9‐ylidene)thiosemicarbazide, C₁₄H₁₂N₄S, (III), with dimethyl acetylenedicarboxylate and methyl propiolate, respectively. The crystal structures of (I), (II) and (III) are molecular and can be considered in two parts: (i) the nearly planar acridine moiety and (ii) the singular heterocyclic ring portion [thiazolidine for (I) and thiazine for (II)] including the linking amine and imine N atoms and the methyl C atom, or the full side chain in the case of (III). The structures of (I) and (II) are stabilized by N—H...O hydrogen bonds and different π–π interactions between acridine moieties and thiazolidine and thiazine rings, respectively.