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An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth g≥7 is equitably Δ(G)-colorable, where Δ(G) is the maximum degree of G.

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]

Electrochemiluminescence of the luminol-O2 system in an electrolyte-free N,N-dimethylformamide (DMF)-dipropylamine (DPA) cosolution is induced by the formation of a carbamate ionic liquid (IL) from the reaction between CO2 and DPA, on the basis of which a facile ECL sensor for measuring atmospheric CO2 has been developed. This ECL sensing method shows several advantages in the detection of CO2, such as high safety, high selectivity, wide linear response range, and good sensitivity. The gas sensor was found to have a linear response range from 100 ppm to 100 v/v% and a detection limit of 80 ppm (at signal-to-noise ratio of 3). This is the first reported IL-induced ECL sensor for a gas, thus the principle of this type of sensor and the IL-induced ECL mechanism have been demonstrated in detail. [ABSTRACT FROM AUTHOR]

It is known that the bandwidth sum of a hypergraph, over all the labellings of the vertices with distinct integers, is the minimum in the total sum of the maximum difference between the labels of the vertices in one edge. Let H be a hypergraph. Denote bandwidth sum of H by BS(H) and the bandwidth sum of a graph G by BS(G). In this paper, the relationship between BS(H) and BS(G) is given. [ABSTRACT FROM AUTHOR]