A note on the transmission feasibility problem in networks

In the networking designing phase, the network needs to be built according to certain indicators to ensure that the network has the ideal functions and can work smoothly. From a modeling perspective, each site in the network is represented by a vertex, channels between sites are represented by edges, and thus the entire network can be denoted as a graph. Problems in the network can be transformed into corresponding graph problems. In particular, the feasibility of data transmission can be transformed into the existence of fractional factors in network graph. This note gives an independent set neighborhood union condition for the existence of fractional factors in a special setting, and shows that the neighborhood union condition is sharp.

A neigborhood union condition for nonadjacent vertices in graphs

A simple graph [Formula: see text] is called [Formula: see text]-bounded if for every two nonadjacent vertices [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the set of neighbors of the vertex [Formula: see text] in [Formula: see text]. In this paper, some properties of [Formula: see text]-bounded graphs are studied. It is shown that any bipartite [Formula: see text]-bounded graph is a complete bipartite graph with at most two horns; in particular, any [Formula: see text]-bounded tree is either a star or a two-star graph. Also, we prove that any non-end vertex of every [Formula: see text]-bounded graph is contained in either a triangle or a rectangle. Among other results, it is shown that all regular [Formula: see text]-bounded graphs are strongly regular graphs. Finally, we determine that how many edges can an [Formula: see text]-bounded graph have?

Litwa w polskiej polityce zagranicznej w latach 2007–2014

Lithuania in Polish Foreign Policy in the Years 2007–2014Lithuania plays a crucial part in the Polish eastern policy. Poland shares a complicated past with its north-eastern neighbour as well as many common interests. One can call it “a love-hate relationship”. The main purpose of this article is to investigate Polish-Lithuanian relations in four main areas: energy recourses and transmission lines; security and democratisation in the Eastern Europe; cooperation within the European Union; condition of minorities. The article focuses on ears 2007–2014 when Donald Tusk was the Prime Minister of Poland and the Ministry of Foreign Affairs was led by Radosław Sikorski. The paper also presents the role of the Polish president and parliament.

Sprouting and Rooting Evaluation of 32 Mulberry Collections under La Union Condition

Mulberry (Morus sp.; Moraceae) is a perennial tree crop, which provides sustainable economic and environmental benefits. Its foliage is the only natural feed available for the silkworm Bombyx mori Linneaus for the production of silk. The mulberry germplasm of the Sericulture Research and Development Institute, Don Mariano Marcos Memorial State University (DMMMSU-SRDI), maintains different exotic and indigenous collections. This study aimed to select breeding materials with high propagation traits for crop improvement and to identify suitable parent materials. Pooled data of 2009 and 2010 evaluations on sprouting, rooting and root proliferation rate at 20, 60 and 90DAP, respectively, were subjected to Analysis of Variance. Selection of variety was done using the Least Significant Difference (LSD) using Batac a widely used a variety for cocoon production, as check variety. Result revealed that there was no significant variation among the 32 mulberry collections as to the number and weight of roots and root-shoot ratio by length and weight. Identified parent materials for improved rooting capacity are S3, Papua, King, MA, Collection2 and H4, thus, considered for drought resistance. Keywords - Plant ecology, mulberry germplasm, propagation traits, rooting and sprouting evaluation, La Union, Philippines

Color Neighborhood Union Conditions for Long Heterochromatic Paths in Edge-Colored Graphs

Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lfloor{2s+4\over5}\rfloor$. In the present paper, we prove that $G$ has a heterochromatic path of length at least $\lceil{s+1\over2}\rceil$, and give examples to show that the lower bound is best possible in some sense.