A Simpler Proof of the Excluded Minor Theorem for Higher Surfaces

We give a simple proof of the fact (which follows from the Robertson–Seymour theory) that a graph which is minimal of genusgcannot contain a subdivision of a large grid. Combining this with the tree-width theorem and the quasi-wellordering of graphs of bounded tree-width in the Robertson–Seymour theory, we obtain a simpler proof of the generalized Kuratowski theorem for each fixed surface. The proof requires no previous knowledge of graph embeddings.     

Reduction of effects of structured non-specific absorption in the determination of arsenic and selenium by electrothermal atomic absorption spectrometry

The presence of iron and phosphates in biological matrices causes deuterium arc background-correction systems to overcompensate at several arsenic and selenium resonance lines. The addition of platinum as matrix modifier has a significant effect on both the absorbance/time profile of iron and the formation of gaseous phosphate decomposition products. A nickel/platinum matrix modifier is shown effectively to control the problems in the determination of selenium arising both from thermal instability and spectral interferences. The same combination eliminates the spectral interferences found at the arsenic resonance lines. Remaining problems are the thermal stabilization of organometallic arsenic compounds present in biological samples. When radioactived-labelled ⁷⁴As compounds prepared in vivo were applied, none of the tested matrix modifiers (Ni, Cu, Ag, Pd, Zr, Ce, Ce + magnesium nitrate) showed a significant influence on the volatility of arsenic in whole blood and urine from rats.     

Stability of workroom air volatile organic compounds on solid adsorbents for thermal desorption gas chromatography

The storage stability of the occupationally frequently occurring compounds, methylethylketone, methylisobutylketone, benzene, toluene, tetrachloroethylene, n-butylacetate, -pinene, β-pinene, limonene and n-decane, has been investigated on the adsorbents Tenax TA, Chromosorb 106 and Carbotrap using thermally desorbable tube type samplers, commonly utilized in ambient and workroom atmospheric measurements. Fifty and 500 ng of each compound were loaded on the various adsorbents tubes, stored at both ambient (20 °C) and refrigerated (4 °C) temperatures and analysed by means of thermal gas chromatography with mass spectrometric detection on days 0, 7, 14 and 28 after exposure. A 90% storage recovery was chosen as acceptance criteria for storage stability, and statistical testing by Student's t-test, analysis of variance and Bonferroni post hoc tests were employed to investigate the effect of the categorical variables storage time, storage temperature and analyte loading on the different adsorbents. Chromosorb 106 showed the overall best behaviour with recoveries of 90% or better for all analytes during the 28-day test period. Tenax TA and Carbotrap yielded lower recoveries and were more influenced by variations in storage time, storage temperature and analyte loading. Refrigerated temperatures were best avoided for storage on Tenax TA, but may increase the recovery of some compounds on Carbotrap (e.g. n-butylacetate). The blank build-up on the adsorbents was also investigated, and Carbotrap and Tenax TA showed no signs of artefact development over time. Chromosorb 106, however, contained inherently more artefacts that build up over time, which in spite of the excellent storage capability, may limit its use in field studies where long storage times are normal.     

1,1,2,2-tetrabromodisilane: gas-phase molecular structure and conformational composition as determined by electron diffraction

The molecular structure of 1,1,2,2-tetrabromodisilane has been investigated using gas-phase electron diffraction data obtained at 110°C. At this temperature the molecules exist as a mixture of about equal parts (X = 0.5 ±0.2) of the two conformers with the H---Si---Si---H torsion angle equal to 180° (anti) or 60° (gauche). Assuming that the two conformers differ in their geometries only in the torsion angle φ, some of the important distance (r_a) and angle () parameters are: r(Si---Si) = 2.349(19) Å, r(Si---Br) = 2.205(5) Å, r(Si---H) = 1.485 Å (assumed), Br---Si---Br = 110.1(1.6)°, Si---Si---Br = 107.1(1.2)° Si---Si---H = 108.6° (assumed). The error limits are 2σ. The observed conformational composition (X_anti = 0.5(0.2)) corresponds to an energy difference between the conformers of ΔE = E(gauche) — E(anti) = 0.5 ± 0.6 kcal mol⁻¹, assuming ΔS = Rln2.     

3-connected planar spaces uniquely embed in the sphere

We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in has a unique accumulation point in , then has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.

     

Planar and infinite hypohamiltonian and hypotraceable graphs

Chvátal raised the question whether or not planar hypohamiltonian graphs exist and Grünbaum conjectured the nonexistence of such graphs. We shall describe an infinite class of planar hypohamiltonian graphs and infinite classes of planar hypotraceable graphs of connectivity two (resp. three). Infinite hypohamiltonian (resp. htpotraceable) graphs are also described. It is shown how the study of infinite hypotraceable graphs leads to a new infinite family of finite hypotraceable graphs.     

Hamiltonian-connected tournaments

We characterize weakly Hamiltonian-connected tournaments and weakly panconnected tournaments completely and we apply these results to cycles and bypasses in tournaments with given irregularity, in particular, in regular and almost regular tournaments. We give a sufficient condition in terms of local and global connectivity for a Hamiltonian path with prescribed initial and terminal vertex. From this result we deduce that every 4-connected tournament is strongly Hamiltonian-connected and that every edge of a 3-connected tournament is contained in a Hamiltonian cycle of the tournament and we describe infinite families of tournaments demonstrating that these results are best possible.     

Bidirectional retracting-free double tracings and upper embeddability of graphs

We characterize those finite connected graphs which admit a closed walk such that each edge is traversed once in each direction and such that no edge is succeeded by the same edge in the opposite direction. This solves an old problem of O. Ore.     

On the Minimum Number of Spanning Trees in k‐Edge‐Connected Graphs

We show that a k‐edge‐connected graph on n vertices has at least spanning trees. This bound is tight if k is even and the extremal graph is the n‐cycle with edge multiplicities . For k odd, however, there is a lower bound , where . Specifically, and . Not surprisingly, c₃ is smaller than the corresponding number for 4‐edge‐connected graphs. Examples show that . However, we have no examples of 5‐edge‐connected graphs with fewer spanning trees than the n‐cycle with all edge multiplicities (except one) equal to 3, which is almost 6‐regular. We have no examples of 5‐regular 5‐edge‐connected graphs with fewer than spanning trees, which is more than the corresponding number for 6‐regular 6‐edge‐connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.