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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]

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 ... 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]