A generalization of chromatic index

AbstractLet G = (V, E) be a graph and k ⩾ 2 an integer. The general chromatic index χ′k(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph 〈F〉 induced sby F has size at most k - 1. This paper initiates a study of χ′k(G) and generalizes some known results on chromatic index

POM2/CSI1 is essential for functional association of cellulose synthase and microtubules in Arabidopsis

In plants, regulation of cellulose synthesis is fundamental for morphogenesis and plant growth. Cellulose is synthesized at the plasma membrane, and the orientation of synthesis is guided by cortical microtubules; however, the guiding mechanism is currently unknown. We show that the conditional root elongation pom2 mutants are impaired in cell elongation, fertility, and microtubule-related functions. Map-based cloning of the POM-POM2 locus revealed that it is allelic to CELLULOSE SYNTHASE INTERACTING1 (CSI1). Fluorescently tagged POM2/CSI1s associated with both plasma membrane-located cellulose synthases (CESAs) and post-Golgi CESA-containing compartments. Interestingly, while CESA insertions coincided with cortical microtubules in the pom2/csi1 mutants, the microtubule-defined movement of the CESAs was significantly reduced in the mutant. We propose that POM2/CSI1 provides a scaffold between the CESAs and cortical microtubules that guide cellulose synthesis

Tropical Dominating Sets in Vertex-Coloured Graphs

Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give upper bounds for random graphs. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure

When the vertex coloring of a graph is an edge coloring of its line graph - a rare coincidence

The 3-consecutive vertex coloring number psi(3c)(G) of a graph G is the maximum number of colors permitted in a coloring of the vertices of G such that the middle vertex of any path P-3 subset of G has the same color as one of the ends of that P-3. This coloring constraint exactly means that no P-3 subgraph of G is properly colored in the classical sense. The 3-consecutive edge coloring number psi(3c)'(G) is the maximum number of colors permitted in a coloring of the edges of G such that the middle edge of any sequence of three edges (in a path P-4 or cycle C-3) has the same color as one of the other two edges. For graphs G of minimum degree at least 2, denoting by L(G) the line graph of G, we prove that there is a bijection between the 3-consecutive vertex colorings of G and the 3-consecutive edge colorings of L(G), which keeps the number of colors unchanged, too. This implies that psi(3c)(G) = psi(3c)'(L(G)); i.e., the situation is just the opposite of what one would expect for first sight

Monoclonal antibody-conjugated dendritic nanostructures for siRNA delivery

Small interfering RNA (siRNA) is a promising tool for gene therapy-based disease treatments. However, delivery of siRNA to the target cells requires a specific and reliable carrier system. Herein we describe a targeted carrier system that can deliver siRNA to cancer cells overexpressing the human epidermal growth factor 2 (HER2) receptor. Trastuzumab-conjugated poly(amido)amine dendrimers can be synthesized using the protocols described here

DEFECTIVE KERNEL1 regulates cellulose synthesis and affects primary cell wall mechanics

The cell wall is one of the defining features of plants, controlling cell shape, regulating growth dynamics and hydraulic conductivity, as well as mediating plants interactions with both the external and internal environments. Here we report that a putative mechanosensitive Cys-protease DEFECTIVE KERNEL1 (DEK1) influences the mechanical properties of primary cell walls and regulation of cellulose synthesis. Our results indicate that DEK1 is an important regulator of cellulose synthesis in epidermal tissue of Arabidopsis thaliana cotyledons during early post-embryonic development. DEK1 is involved in regulation of cellulose synthase complexes (CSCs) by modifying their biosynthetic properties, possibly through interactions with various cellulose synthase regulatory proteins. Mechanical properties of the primary cell wall are altered in DEK1 modulated lines with DEK1 affecting both cell wall stiffness and the thickness of the cellulose microfibril bundles in epidermal cell walls of cotyledons