An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.
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An arc-coloured digraph D = (V,A) is said to be rainbow connected if for every pair {u, v} ⊆ V there is a directed uv-path all whose arcs have different colours and a directed vu-path all whose arcs have different colours. The minimum number of colours required to make the digraph D rainbow connected is called the rainbow connection number of D, denoted rc⃗ (D). A cactus is a digraph where each arc belongs to exactly one directed cycle. In this paper we give sharp upper and lower bounds for the rainbow connection number of a cactus and characterize those cacti whose rainbow connection number is equal to any of those bounds. Also, we calculate the rainbow con- nection numbers of some infinite digraphs and graphs, and present, for each n ≥ 6, a tournament of order n and rainbow connection number equal to 2.
In this study, the effectiveness of using natural bio-coagulants and bio-flocculants to treat textile wastewater through the coagulation-flocculation method was examined. These bio-based agents have several advantages over chemical agents, including biodegradability, natural abundance, low toxicity, and low cost. A bio-coagulant (holm oak acorn (HOA)) and a bio-flocculant (cactus juice) were used to investigate the capacity for turbidity removal and decolorization of textile wastewater. The UV spectrophotometer was used to characterize the discharges before and after treatment, and the chemical oxygen demand (COD) and biological oxygen demand (BOD5) levels were calculated. Box-Behnken design (BBD) coupled with response surface methodology (RSM) were utilized to optimize the process and reduce turbidity and decolorization in textile wastewater. The obtained results show that under the optimal conditions (0.5 g·L-1 of HOA, 15 mL·L-1 of cactus juice, and a pH of 7), decolorization and turbidity removal were achieved at 69% and 90%, respectively. This study demonstrates the potential of using bio-coagulants and bio-flocculants in the treatment of textile wastewater.
Representatives of the family Cactaceae are characterized by a wide range of phyllotaxis. To assess the origin of this diversity, early stages of phyllotactic pattern formation were examined in seedlings. The analysis of the sequence of areole initiation revealed intertribal differences. In seedlings from the Trichocereeae (Gymnocalycium, Rebutia) and Notocacteae (Parodia) tribes, two opposite cotyledonal areoles developed as the first elements of a pattern. Usually, next pair of areoles was initiated perpendicularly to cotyledonal areoles, starting the decussate pattern. This pattern was subsequently transformed into bijugate or into simple spiral phyllotaxis. In seedlings from the Cacteae tribe (Mammillaria and Thelocactus), cotyledonal areoles were never observed and the first areoles always appeared in the space between cotyledons. It was either areole pair (mainly in Mammillaria), starting a decussate pattern, or a single areole (mainly in Thelocactus) quickly followed by areoles spirally arranged, usually in accordance with the main Fibonacci phyllotaxis. Differences in the initial stages of pattern formation do not fully explain the phyllotaxis diversity in mature cacti. Only two, the most common phyllotactic patterns occurred in the early development of studied seedlings, i.e. the main Fibonacci and the decussate pattern. Discrepancy in the range of phyllotactic spectra in seedlings and in mature plants suggests that phyllotaxis diversity emerges during further plant growth. Initial phyllotactic transformations, occurring already in the very early stages, indicate great plasticity of cactus growth and seem to support the hypothesis of the ontogenetic increase of phyllotaxis diversity due to transformations.
In this paper we determine, or give lower and upper bounds on, the 2-dipath and oriented L(2, 1)-span of the family of planar graphs, planar graphs with girth 5, 11, 16, partial k-trees, outerplanar graphs and cacti.
Badano wpływ jakości światła i ciemności na tempo regeneracji kalusa embriogennego Gymnocalycium mihanovichii (FRIČ ET GÜRKE) BRITTON ET ROSE f. aurantiaca in vitro. Stwierdzono, że przyrost kalusa na zmodyfikowanej pożywce MS bez regulatorów wzrostu był największy pod światłem żółtym oraz czerwonym. Największa zaś liczba kalusów zregenerowała struktury embrioidalne pod światłem żółtym, dziennym oraz czerwonym.
EN
The influence of light quality and darkness on the rate of regeneration of callus and embryogenic structures of Gymnocalycium mihanovichii (FRIČ ET GÜRKE) BRITTON ET ROSE f. aurantiaca in vitro was studied. The increment of callus on the modified MS medium without growth regulators was biggest under the yellow and red light conditions, and a high number of calli regenerated embryogenic structures under the yellow, daylight and red light conditions.