Ab initio calculations at HF/6-31G* level of theory for geometry optimization and MP2/6-31G*//HF/6-31G* for a single point total energy calculation are reported for the importantenergy-minimum conformations and transition-state geometries of 1, 3-diazacyclohepta-1, 2-diene (2) and 1, 3-diazacycloocta-1, 2-diene (3). The C2 symmetric twist-chair (2-TC)conformation of 2 is calculated to be 7.4 kJ...

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph Kn that has no rainbow spanning subgraph with diameter at most t. We prove s(n, t) = (n−2 2 ) +1 for n, t ≥ 3, while s(n, 2) = (n−2 2 )

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph G and a positive integer n, the anti-Ramsey number ar(n, G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H. Anti-Ramsey numbers were introduced by Erdős, Simonovits and Sós and studied in numerous papers. Let G be a graph with anti-Ramsey number ar(n,...

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G , denoted by rxk(G). ...

Let G be an edge-colored copy of Kn, where each color appears on at most n/2 edges (the edgecoloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth [4] conjectured that every properly edge-colored Kn (n ≥ 6 and even) using exactly n−1 colors has n/2 edge-disjoint rainbow spanning trees, and they proved...

An edge-colored graph G is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rcðGÞ, is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rcðGÞ is an NP-hard problem, as well as that even deciding whether a graph has rcðGÞ...

The Rainbow Ramsey Theorem is essentially an “anti-Ramsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ra...

A 9-week feeding trial was conducted to evaluate the effects of dietary cholesterol supplementation at different levels (0, 0·3, 0·6, 0·9, 1·2 and 1·5 %) on growth and cholesterol metabolism of rainbow trout (Oncorhynchus mykiss) fed soyabean meal (SBM)-based diets. Daily growth coefficient (DGC) steadily increased when the supplemental cholesterol was increased by up to 1·2 %, but declined upo...