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For two graphs G and H, the Gallai–Ramsey number gr k (G : H) is defined as the minimum integer n such that any k-edge-coloring of K n must contain either a rainbow copy of G or a monochromatic copy of H. In this paper, we obtain the exact values of gr k (G : H) , where H is a path and G ∈ { K 1 , 3 , P 4 + , P 5 } is a small tree and P 4 + is the graph consisting of P 4 with one extra edge incident with an inner vertex. [ABSTRACT FROM AUTHOR]

Combinatorics of biopolymer structures, especially enumeration of various RNA secondary structures and protein contact maps, is of significant interest for communities of both combinatorics and computational biology. However, most of the previous combinatorial enumeration results for these structures are presented in terms of generating functions, and few are explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas for a particular class of biologically relevant structures, say, saturated 2-regular simple stacks, whose configuration is related to protein folds in the 2D honeycomb lattice. We establish a semi-bijective algorithm that converts saturated 2-regular simple stacks into forests of small trees, which produces a uniform formula for saturated extended 2-regular simple stacks with any of the six primary component types. Summarizing the six different primary component types, we obtain a bivariate explicit formula for saturated extended 2-regular simple stacks with n vertices and k arcs. As consequences, the uniform formula can be reduced to Clote's results on k -saturated 2-regular simple stacks and the optimal 2-regular simple stacks, and Guo et al.'s result on the optimal extended 2-regular simple stacks. [ABSTRACT FROM AUTHOR]

Stand structure and woody species diversity in a subtropical evergreen broadleaf forest grown in a silicate habitat, Okinawa Island, have been investigated on the basis of stand stratification. The forest stand consisted of four layers. The floristic composition of the top and the lower three layers was only slightly similar, although approximately one-third of the species were common to them. Mean tree weight decreased from the top toward the bottom layer whereas tree density increased from the top downward. This trend resembled the mean weight–density trajectory of self-thinning plant populations. The relationship between mean tree height and tree density for the upper two layers supported Yamakura’s quasi −1/2 power law of tree height. The values of the Shannon–Wiener index, H′, and the equitability index, J′, tended to increase from the top layer downward except for the bottom layer. The values of H′ and J′ were, respectively, 4.83 bit and 0.82 for trees taller than 0.10 m. The lower layers contained many species of smaller height. High species diversity of the forest depended on small trees in the lower layers. Conservation of small trees in the lower layers, especially the bottom layer, is indispensable for sound maintenance of Okinawan evergreen broadleaf forests. [ABSTRACT FROM AUTHOR]