Matchings in random spanning subgraphs of cubelike graphs

A question about the evolution of random spanning subgraphs G _p of bipartite regular so called cubelike graphs G is considered. It is shown that for G _p of any large enough cubelike graph G the threshold to have a 1-factor is the same as the threshold to have no isolated vertices. This generalizes a conjecture of K. Weber.     

The total coloring of a multigraph with maximal degree 4

In this paper it is proved that every multigraph with maximal degree 4 has a total coloring in six colors.     

On the chromatic number of set systems

An (r, l)‐system is an r‐uniform hypergraph in which every set of l vertices lies in at most one edge. Let m_k(r, l) be the minimum number of edges in an (r, l)‐system that is not k‐colorable. Using probabilistic techniques, we prove that where b_{r, l} is explicitly defined and a_{r, l} is sufficiently small. We also give a different argument proving (for even k) where a′_{r, l}=(r?l+1)/r(2^r?1re)^?l/(l?1). Our results complement earlier results of Erd?s and Lovász [10] who mainly focused on the case l=2, k fixed, and r large. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 87–98, 2001     

Total choosability of multicircuits I

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch′′(C) equal to its ordinary total chromatic number χ′′(C). In the present paper, the kernel method is used to prove this for every multicircuit that has at least two vertices with degree less than its maximum degree Δ. The result is also proved for every multicircuit C for which χ′′(C)≥Δ+2. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 26–43, 2002     

Total choosability of multicircuits II

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this paper, the method of Alon and Tarsi is used to prove that all multicircuits of even order, and some regular and near‐regular multicircuits of odd order have total choosability (i.e., list total chromatic number) equal to their ordinary total chromatic number. This completes the proof that every multicircuit has total choosability equal to its total chromatic number. In the process, the total chromatic numbers of all multicircuits are determined. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 44–67, 2002     

Synthesis of 3-(2-Aryloxazolidin-3-yl)tropanes

We obtained 3-(2-aryloxazolidin-3-yl)tropanes by reaction of aromatic aldehydes with 3-[N-(2-hydroxyethyl)amino]tropane. We obtained 3-benzyloxazolidine-2-spiro-3'-(8'-carbethoxy)nortropane by reaction of benzylaminoethanol with 8-carbethoxynortropan-3-one.     

Electrical transport and magnetic ordering in R 2Ti3Ge4 (R=Dy, Ho and Er) compounds

New R ₂Ti₃Ge₄ (R=Dy, Ho and Er) intermetallic compounds have been synthesized and characterized by X-ray diffraction and low temperature ac magnetic susceptibility, electrical resistivity and thermoelectric power measurements were carried out. The compounds crystallize in the parent, Sm₅Ge₄-type orthorhombic structure (space group Pnma) and lanthanide contraction is observed as one moves along the rare-earth series. The changeover from paramagnetic to antiferromagnetic phase happens at low temperatures and the ordering temperature scales with the de Gennes factor. The electrical resistivity is metallic with a negative curvature above 100 K. Thermopower displays a weak maximum at temperatures less than 50 K signifying the possible phonon and magnon drag effects.     

Acyclic and oriented chromatic numbers of graphs

The oriented chromatic number χ_o(G ^→) of an oriented graph G ^→ = (V, A) is the minimum number of vertices in an oriented graph H ^→ for which there exists a homomorphism of G ^→ to H ^→. The oriented chromatic number χ_o(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for χ_o(G) in terms of χ_a(G). An upper bound for χ_o(G) in terms of χ_a(G) was given by Raspaud and Sopena. We also give an upper bound for χ_o(G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. © 1997 John Wiley & Sons, Inc.