Edges in graphs with large girth

Several upper bounds are given for the maximum number of edgese possible in a graph depending upon its orderp, girthg and, in certain cases, minimum degree. In particular, one upper bound has an asymptotic order ofp ^1+2/(g–1) wheng is odd. A corollary of our final result is that whenk = e/p 2. Asymptotic and numerical comparisons are also presented.     

Modeling mechanical properties of polyhydroxyalkanoate during degradation in animal tissue

Polyhydroxyalkanoates (PHAs) are a family of biodegradable and biocompatible polymers produced by several species microorganisms that possess favorable mechanical properties (e.g. strength and elongation properties). Different types of PHA polymers have been used in medical applications. However, in order to better understand the use of this polymer in the different applications, a thorough understanding of the kinetics of in vivo degradation is one of the major requirements. In this study, poly(3‐hydroxybutyrate) (PHB) was subcutaneously implanted in mice and incubated for 2, 4, 8, or 16 weeks. After removal from the animal, the strength, elongation, mass loss, and enthalpy of the PHB were tested for each time point. From these data, a mathematical model was generated by Rayleigh's method of dimensional analysis, where polymer strength over tissue contact time could be predicted. To prove the model, previous data obtained by our group were used: poly(3‐hydroxybutyrate‐co‐3‐hydroxyhexanoate) [P(HB‐co‐HHx)] incubation in the presence of human embryonic kidney cells (HEK). It was found that the developed model was aligned with experimental results, could predict the strength of the polymer when in contact with cells, and the predicted strength follows the trend of the experimental data. Also, the dimensionless constant (K) value associated with the model is different for both experiments, where this constant, produced via experimental data, is used for construction of a homogeneous equation. Copyright © 2017 John Wiley & Sons, Ltd.     

In situ vinylindole synthesis. Diels-alder reactions with maleimides to give tetrahydrocarbazoles

Tetrahydrocarbazoles have been prepared in one-flask syntheses from indoles, ketones or aldehydes, and maleimides, with acid catalysis. The reactions involve a condensation of the indole with the ketone or aldehyde, followed by an in situ trapping of the vinylindole in a Diels-Alder addition with a maleimide. Isomerization of the double bond into the indole nucleus gave the tetrahydrocarbazoles which were isolated ( 6, 9 , and 10 ). Variation of the indole, carbonyl compound, and maleimide has been explored. The predominant stereochemistry of the tetrahydro ring in the products is all-cis, although a second stereoisomer has been isolated. Two regioisomers were generated from all unsymmetrical 2-alkanones, except 2-butanone, which gave the single isomer 9a . Aromatization of tetrahydrocarbazoles 6 to carbazoles 7 was accomplished with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone.     

Generalized k-tuple colorings of cycles and other graphs

Stahl's concept of k-tuple coloring the nodes of a graph is extended to specify that adjacent nodes must have i colors in common. Complete results for a minimum such coloring are obtained for bipartite graphs and odd cycles. Partial results are shown for complete graphs.     

A characterization of competition graphs

Characterization of competition graphs for arbitrary and acyclic directed graphs are presented.     

Graphs which,with their complements,have certain clique covering numbers

Two common invariants of a graph G are its node clique cover number, θ₀(G), and its edge clique cover number, θ₁(G). We present in this work a characterization of those graphs for which they and their complements, $\text{G}\text{?}$, have θ₀(G)=θ₁(G) and θ₀($\text{G}\text{?}$)=θ₁($\text{G}\text{?}$). Graphs satis ying these conditions are shown to constitute a subset of those graphs which we term C-graphs.     

On the dimension of trees

We consider isometric embedding of trees into the infinite graph Z_m whose vertices are the m-dimensional lattice points where two vertices a=(a₁,a₂,…,a_m) and b=(b₁,b₂,…,b_m) are adjacent if and only if |a_i-b_i|?1 for 1?i?m. Linial, London, and Rabinovich have shown that this can be done with , where t is the number of leaves. In this note, we sketch a proof that .     

Powerful alliances in graphs

For a graph G=(V,E), a non-empty set S⊆V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V−S than it has in S, and S is an offensive alliance if for every v∈V−S that has a neighbor in S, v has more neighbors in S than in V−S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.     

Security in graphs

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. Necessary and sufficient conditions for a set to be secure are determined.