Synthesis of new derivatives of 5,6,7,8-tetrahydro-1,6-naphthyridines and their pharmacological properties

1,5-Dicarbonyl derivatives of piperidine and tropane were studied in the heterocyclization with hydroxylamine hydrochloride. All the 5,6,7,8-tetrahydro-1,6-naphthyridines obtained exhibit a pronounced analgesic effect.__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 251–253, January, 2005.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

The dimension of suborders of the Boolean lattice

We consider the order dimension of suborders of the Boolean latticeB _n . In particular we show that the suborder consisting of the middle two levels ofB _n dimension at most of 6 log₃ n. More generally, we show that the suborder consisting of levelss ands+k ofB _n has dimensionO(k ² logn).The research of the second author was supported by Office of Naval Research Grant N00014-90-J-1206.The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

Packing and Covering Triangles in K 4-free Planar Graphs

We show that every K ₄-free planar graph with at most ν edge-disjoint triangles contains a set of at most ${\frac32\nu}$ edges whose removal makes the graph triangle-free. Moreover, equality is attained only when G is the edge-disjoint union of 5-wheels plus possibly some edges that are not in triangles. We also show that the same statement is true if instead of planar graphs we consider the class of graphs in which each edge belongs to at most two triangles. In contrast, it is known that for any c?<?2 there are K ₄-free graphs with at most ν edge-disjoint triangles that need more than cν edges to cover all triangles.     

Dense uniform hypergraphs have high list chromatic number

The first author showed that the list chromatic number of every graph with average degree $d$ is at least $(0.5?o\left(1\right)){log}_{2}d$. We prove that for $r\ge 3$, every $r$-uniform hypergraph in which at least half of the $(r?1)$-vertex subsets are contained in at least $d$ edges has list chromatic number at least $\frac{lnd}{100r^3}$. When $r$ is fixed, this is sharp up to a constant factor.