On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

Monodentate sigma-N and Bidentate sigma-N,sigma-N' Coordination of 1,1-Bis((N-p-tolylimino)diphenylphosphoranyl)ethane, CHCH(3)(PPh(2)=NC(6)H(4)-4-CH(3))(2), to Platinum(II)

The new ligand, 1,1-bis((N-p-tolylimino)diphenylphosphoranyl)ethane (1,1-BIPE), 1, has been synthesized by means of a Staudinger reaction of 1,1-bis(diphenylphosphino)ethane (1,1-dppe) with 2 equiv of p-tolylazide. Bridge-splitting reactions of Pt(2)Cl(4)(PR(3))(2) with 1 readily afforded sigma-N monodentate complexes, [PtCl(2)(PR(3)){1,1-BIPE-sigmaN}] (2a, PR(3) = PEt(3); 2b, PR(3) = PMe(2)Ph). Conversion of 2 into the six-membered platinacycle [PtCl(PR(3)){1,1-BIPE-sigmaN,sigmaN'}](+)[X](-) (3) (X = Cl, PtCl(3)(PR(3)), BF(4)) took place after prolonged stirring, its reaction rate being strongly dependent on the type of phosphine (>5 days for 2ain the presence of NaBF(4), 1 h for 2b) and the metal-to-ligand ratio. The compounds 1, 2, and 3 have been fully characterized by (1)H, (31)P{(1)H}, and (13)C{(1)H} NMR and IR spectroscopy, elemental analysis, or FAB mass spectroscopy. The molecular structures of CHCH(3)(PPh(2)=NC(6)H(4)-4-CH(3))(2) (1) and [PtCl(PMe(2)Ph){(N(pTol)=PPh(2))(2)CHCH(3)}](+)[Cl](-) (3b) have been determined by X-ray crystallography. Crystal data for 1: space group P2(1)/c with a = 8.9591(5) ?, b = 19.1961(12) ?, c = 21.9740(9) ?, beta = 105.069(4) degrees, V = 3649.1(3) ?(3), and Z = 4. The structure refinement converged to R = 0.080 and R(w) = 0.109. Crystal data for 3b: monoclinic, space group P2(1)/c with a = 12.4021(7) ?, b = 16.9705(11) ?, c = 23.760(2) ?, beta = 109.544(5) degrees, V = 4712.7(5) ?(3), and Z = 4. The structure refinement converged to R1 = 0.057, wR2 = 0.122. Variable temperature NMR spectroscopy has revealed that complexes 3 exclusively adopt a twisted boat conformation with the methyl group in equatorial position at low temperature, in agreement with the solid state structure of 3b as determined by X-ray crystallography. Boat-to-boat inversion is assumed to take place at temperatures above 293 K. Furthermore, for 3, hindered rotation of one of the p-tolyl substituents on nitrogen has been established at low temperatures.     

On the solitaire cone and its relationship to multi-commodity flows

The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give:?1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone;?2. a related NP-completeness result;?3. a method of generating large classes of facets;?4. a complete characterization of 0-1 facets;?5. exponential upper and lower bounds (in the dimension) on the number of facets;?6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards;?7. a complete characterization of the adjacency of extreme rays, diameter, number of 2-faces and edge connectivity for rectangular toric boards. Received: July 1996 / Accepted: February 2000?Published online February 22, 2001     

On the $${\mathcal {}}$$-free extension complexity of the TSP

It is known that the extension complexity of the TSP polytope for the complete graph \(K_n\) is exponential in n even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets \({\mathcal {H}}\) of facet-defining inequalities of the TSP polytope. In particular, we consider the case when \({\mathcal {H}}\) is either the set of blossom inequalities or the simple comb inequalities. These inequalities are routinely used in cutting plane algorithms for the TSP. We show that the extension complexity remains exponential even if we exclude these inequalities. In addition we show that the extension complexity of polytope formed by all comb inequalities is exponential. For our proofs, we introduce a subclass of comb inequalities, called (h, t)-uniform inequalities, which may be of independent interest.     

One-photon photophysics and two-photon absorption of 4-[9,9-di(2-ethylhexyl)-7-diphenylaminofluoren-2-yl]-2,2':6',2'-terpyridine and their platinum chloride complexes

The synthesis, one-photon photophysics and two-photon absorption (2PA) of three dipolar D-π-A 4-[9,9-di(2-ethylhexyl)-7-diphenylaminofluoren-2-yl]-2,2':6',2'-terpyridine and their platinum chloride complexes with different linkers between the donor and acceptor are reported. All ligands exhibit (1)π,π* transition in the UV and (1)π,π*/(1)ICT (intramolecular charge transfer) transition in the visible regions, while the complexes display a lower-energy (1)π,π*/(1)CT (charge transfer) transition in the visible region in addition to the high-energy (1)π,π* transitions. All ligands and the complexes are emissive at room temperature and 77 K, with the emitting excited state assigned as the mixed (1)π,π* and (1)CT states at RT. Transient absorption from the ligands and the complexes were observed. 2PA was investigated for all ligands and complexes. The two-photon absorption cross-sections (σ(2)) of the complexes (600-2000 GM) measured by Z-scan experiment are much larger than those of their corresponding ligands measured by the two-photon induced fluorescence method. The ligand and the complex with the ethynylene linker show much stronger 2PA than those with the vinylene linker.     

Total lattice potential energy of sodium bromide dihydrate NaBr · 2H2O

In addition to presenting comparative calculations by two approaches for the total lattice potential energy of sodium bromide dihydrate, NaBr · 2H₂O, found to take the value 803.9 kJ mol^-1, we investigate the influence of the size and nature of the basis set used to generate multipole moments in a Hartee-Fock calculation which are in turn used to calculate the Madelung constant. The requirement is one of critical size of the basis set and once this is reached the electrostatic energy will be reliable.     

A linear algorithm for computing the visibility polygon from a point

The hidden-line problem in two dimensions is a recurrent problem in computer graphics. In this paper a linear, and thus optimal, algorithm is exhibited for solving this problem.     

On the complexity of finding the convex hull of a set of points

An n log n lower bound is found for linear decision tree algorithms with integer inputs that either identify the convex hull of a set of points or compute its cardinality.     

Excited-state absorption in a terpyridyl platinum(II) pentynyl complex

The singlet excited-state lifetime of a terpyridyl platinum(II) pentynyl complex was determined to be 268+/-87 ps by fitting femtosecond transient absorption data, the triplet excited-state lifetime was found to be 62 ns by fitting nanosecond transient absorption decay data, and the triplet quantum yield was measured to be 0.16. A ground-state absorption cross section of 2.5 x 10(-19) cm(2) at 532 nm was deduced from UV-vis absorption data. Excited-state absorption cross sections of 3.5 x 10(-17) cm(2) (singlet) and 4.5 x 10(-17) cm(2) (triplet) were obtained by using a five-level dynamic model to fit open-aperture Z scans at picosecond and nanosecond pulse widths and a variety of pulse energies.