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The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈Δ(G)/2⌉ ≤ la(G) ≤ ⌈Δ(G)+1/2 ⌉ for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13. [ABSTRACT FROM AUTHOR]

The well-defined heavy rare-earth ytterbium iodide complex 1 (L2YbI) has been successfully employed as an efficient catalyst for the hydroboration of a wide range of aldehydes and ketones with pinacolborane (HBpin) at room temperature. The protocol requires low catalyst loadings (0.1-0.5 mol %) and proceeds rapidly (>99% conversion in <10 min). Additionally, catalyst 1 shows a good functional group tolerance even toward the hydroxyl and amino moieties and displays chemoselective hydroboration of aldehydes over ketones under mild conditions. [ABSTRACT FROM AUTHOR]

A graph G is called a fractional ID-k-factor-critical graph if after deleting any independent set of G the resulting graph admits a fractional k-factor. In this paper, we prove that for k≥2, G is a fractional ID-k-factor-critical graph if δ(G) ≥n/3+k,σ2(G)≥4n/3,n≥6k-8 The result is best possible in some sense. [ABSTRACT FROM AUTHOR]

The Randic index, harmonic index, sum connectivity index and Zagreb index are vital distance-based topological indices which reveal certain structural features and chemical characteristics of organic molecules. In this report, by means of the molecular structural analysis, we consider the general version of these degree-based indices for special classes of circumcoronene series of benzenoid and Capra-designed planar benzenoid series. The accurate expressions of general degree-based indices for circumcoronene homologous series of benzenoid are presented, and several degree-based indices of Capra-designed planar benzenoid series are determined as well. [ABSTRACT FROM AUTHOR]

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices admits different sets of colors. The minimum number of colors needed for such a coloring of G is denoted by χ'a(G). In this paper, we show that if G is a planar graph with maximum degree Δ ≥ 16, then Δ ≤ χ'a(G) ≤ Δ + 1, and χ'a(G) = Δ+1 if and only if G contains two adjacent vertices of maximum degree. [ABSTRACT FROM AUTHOR]

In theoretical chemistry, the second atom-bond connectivity index was introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this paper, we determine the second atom-bond connectivity index of unilateral polyomino chain and unilateral hexagonal chain. Furthermore, the second ABC indices of V-phenylenic nanotubes and nanotori are presented. [ABSTRACT FROM AUTHOR]

Let D be an acyclic orientation of a simple graph G. An arc is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define dmin(G) to be the minimum number of d(D) over all acyclic orientations D of G. Let Cn denote the cycle on n vertices. The cube Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is the graph defined on the same vertex set of Cn such that any two distinct vertices u and v are adjacent in Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. if and only if their distance in Cn is at most 3. In this paper, we study the structure of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. to determine its minimum number of dependent arcs. [ABSTRACT FROM AUTHOR]

A 1-planar graph is a graph that can be drawn in the Euclidean plane such that each edge crosses at most one edge. An independent crossing (IC)-planar graph is a 1-planar graph satisfying the condition that two pairs of crossing edges have no common end-vertices. It is shown in this paper that every IC-planar graph is 6-choosable. [ABSTRACT FROM AUTHOR]

Let ... be a connected digraph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of .... A firefighter starts to protect vertices. At each time interval, the firefighter protects k vertices not yet on fire. Afterward, the fire spreads to all unprotected neighbors that are heads of some arcs starting from the vertices on fire. Let snk(v) denote the maximum number of vertices in ... that the firefighter can save when a fire breaks out at vertex v. The k-surviving rate ρk(...) of ... is defined as .... In this paper, we consider the k-surviving rate of digraphs. Main results are as follows: (1) if ... is a k-degenerate digraph, then ρk(...)≥1/k+1; (2) if ... is a planar digraph, then ρ2(...)>1/40; (3) if ... is a planar digraph without 4-cycles, then ρ1(...)>1/51. [ABSTRACT FROM AUTHOR]

A plane graph G is entirely κ-colorable if V (G) ∪ E(G) ∪ F(G) can be colored with κ colors such that any two adjacent or incident elements receive different colors. In 1993, Borodin proved that every plane graph G with maximum degree Δ ≥ 12 is entirely (Δ + 2)-colorable. In this paper, we improve this result by showing that every plane graph G with Δ ≥ 9 is entirely (Δ + 2)-colorable. [ABSTRACT FROM AUTHOR]

The organopalladium complex containing ortho-metallated (S)-[1-(dimethylamino)ethyl]naphthalene as a chiral auxiliary has been successfully employed to promote the asymmetric cycloaddition reaction between 3,4-dimethyl-1-phenylarsole and diphenylvinylarsine. In the intramolecular cycloaddition reaction, a pair of separable diastereomeric palladium complexes was obtained in the ratio of 6:1. The chiral naphthylamine auxiliary could be removed chemoselectively from the template by treatment with HCl and subsequently NaI to generate the neutral diiodo complex [(As-As)PdI2]. Treatment of the diiodo complex with KCN gave the enantiomerically pure As-As bidentate ligand in quantitative yield. In contrast to the reported similar P-P and As-P analogues, both arsenic donors in the diiodo complex could be readily eliminated to produce a structurally novel dimetallic complex. [ABSTRACT FROM AUTHOR]