Two complementary dipyrromethane + dipyrromethanemonocarbinol routes to a meso-substituted 5-isocorrole were investigated. While both routes could afford the identical 5-isocorrole, self-condensation of the different dipyrromethanemonocarbinol precursors could potentially lead to a second porphyrinoid of different structure (a porphyrin or a porphodimethene). The two reaction routes were examined to compare the distribution of porphyrinoid products, probe the effect of key reaction parameters on the product distribution, and identify conditions for the efficient preparation of the 5-isocorrole so that its spectral properties and stability toward light and air could be assessed. For each route, a systematic survey of reaction parameters was performed via analytical-scale reactions monitored for the yields of the 5-isocorrole and self-condensation product by HPLC. The two reaction routes were found to afford very different product distributions in accordance with the anticipated nucleophilicity of the dipyrromethane and dipyrromethanemonocarbinol precursors. The most promising reaction condition (InCl(3), 0.32 mM) was performed on a preparative-scale providing the 5-isocorrole in an isolated yield of 31% (102 mg). Spectroscopic analysis was consistent with the 5-isocorrole structure. The stability of the 5-isocorrole in dilute solution upon exposure to light and air was assessed by UV-vis spectroscopy and HPLC and was found to be excellent.

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Given a graph H, a graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question:Determine the minimum even integer σ(H, n) such that every n-term graphic sequence with sum at least σ(H, n) is potentially H-graphic.This problem can be viewed as a "potential" degree sequence relaxation of the (forcible) Turán problems. While the exact value of σ(H, n) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary H. In this paper, we determine σ(H, n) asymptotically for all H, thereby providing an Erdős-Stone-Simonovits-type theorem for the Erdős-Jacobson-Lehel problem.

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex $v$ is the number of different colors on edges incident to $v$. Wang and Li conjectured that for $k\geq 4$, every edge-colored graph with minimum color degree at least $k$ contains a rainbow matching of size at least $\left\lceil k/2 \right\rceil$. We prove the slightly weaker statement that a rainbow matching of size at least $\left\lfloor k/2 \right\rfloor$ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least $\left\lceil k/2 \right\rceil$ that fail to hold only for finitely many exceptions (for each odd $k$).

Let G be a weighted graph in which each vertex initially has weight 1. A total acquisition move transfers all the weight from a vertex u to a neighboring vertex v, under the condition that before the move the weight on v is at least as large as the weight on u. The (total) acquisition number of G, written a t (G), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves.Among connected n-vertex graphs, a t (G) is maximized by trees. The maximum is Θ( √ n lg n) for trees with diameter 4 or 5. It is ⌊(n + 1)/3⌋ for trees with diameter between 6 and 2 3 (n + 1), and it is ⌈(2n − 1 − D)/4⌉ for trees with diameter D when 2 3 (n + 1) ≤ D ≤ n − 1. We characterize trees with acquisition number 1, which permits testing a t (G) ≤ k in time O(n k+2 ) on trees.If G = C 5 , then min{a t (G), a t (G)} = 1. If G has diameter 2, then a t (G) ≤ 32 ln n ln ln n; we conjecture a constant upper bound. Indeed, a t (G) = 1 when G has diameter 2 and no 4-cycle, except for four graphs with acquisition number 2.Deleting one edge of an n-vertex graph cannot increase a t by more than 6.84 √ n, but we construct an n-vertex tree with an edge whose deletion increases it by more than 1 2 √ n. We also obtain multiplicative upper bounds under products.