Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s+ t = χ(G)+1, there is a partition (S, T ) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for line graphs of multigraphs.

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G respectively. Let G×H be the direct product graph of graphs G and H . We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G ×H) = max{α(G)|V (H)|, α(H)|V (G)|} and χ(G×H) = min{χ(G), χ(H)}. AMS Classification: 05C15, 05C69.

Transition-metal-based layered double hydroxides (LDHs) have attracted substantial attention as highly efficient oxygen evolution reaction (OER) catalysts because they are earth-abundant, low-cost, and environmentally friendly materials with favorable adsorption/desorption energies for intermittent reactants. However, the application of these LDHs high-performance electrocatalysts is often hind...

We consider locally finite graphs with vertex set N. A graph G is computable if the edge set is computable and highly computable if the neighborhood function NG (which given v outputs all of its adjacent vertices) is computable. Let χ(G) be the chromatic number of G and χ(G) be the computable chromatic number of G. Bean showed there is a computable graph G with χ(G) = 3 and χ(G) =∞, but if G is...

During DNA replication in Escherichia coli, single-stranded DNA-binding protein (SSB) protects single-stranded DNA from nuclease action and hairpin formation. It is known that the highly conserved C-terminus of SSB contacts the χ subunit of DNA polymerase III. However, there only exists a theoretical model in which the 11 C-terminal amino acids of SSB have been docked onto the surface of χ. In ...

We show that the arguments proposed by Park and Yee against our recent derivation of the statistical entropy of 2D black holes do not apply to the case under consideration PACS: 04.70. Dy, 04.50. +h E-Mail: [email protected] E-Mail: [email protected] In a recent Comment [1] Park and Yee claimed that derivation of the statistical entropy of 2D (two-dimensional) black holes proposed by us in Ref...