Let G be a simple planar graph and let α be a positive integer such that 1 ≤ α ≤ 3 and δ(G) ≥ α + 2. For every pair of edges e1, e2 of G, the graph G− {e1, e2} contains an [α, α + 1]-factor.

When 1 < p < ∞ and p 6= 2 the p-harmonic measure on the boundary of the half plane R+ is not subadditive. In fact, there are finitely many sets E1, E2,...,Eκ on R, of p-harmonic measure zero, such that E1 ∪ E2 ∪ ... ∪ Eκ = R.

In this note we present answers to the open problems posed by Brualdi, Kiernan, Meyer and Schroeder in [Cyclic matching sequencibility of graphs, Australas. J. Combin. 53 (2012), 245–256]. 1 Discussion and response Let G ⊆ Kn be a graph of order n with m edges. The matching number of G is the maximum number of edges in a matching. The matching number of a linear ordering e1, e2, . . . , em of t...

Human E1 is a key player in protein ubiquitination, however the E1 structure is not available. In this paper, we describe the derivation of a human E1 structure using molecular modelling based on the crystal structure of S. cerevisiae E1 and M. Musculus E1. Key interactions between our E1 model and ubiquitin are also discussed.

A long-running debate in the literature on conditioning in humans focuses on the question of whether conditioned responses are the product of automatic link formation processes governed by the standard laws of simple associative learning, or the consequence of participants' inferences about the relationships between the 2 related events, E1 and E2, which would lead E1 to generate a conscious ex...

The chain graph sandwich problem asks: Given a vertex set V , a mandatory edge set E1, and a larger edge set E2, is there a graph G= (V ,E) such that E1 ⊆ E ⊆ E2 with G being a chain graph (i.e., a 2K2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E1 is a connected graph, which has a linear-time solutio...

We report Raman studies of the A1 TO , E1 TO , E2 , A1 LO , and E1 LO symmetry phonons of GaN from 20 to 325 K. By applying anharmonic decay theory to the observed temperature dependences of the phonon energies and linewidths, we determine the phonon decay mechanisms of these zone-center vibrations. Thermal expansion is taken into account using published temperature-dependent coefficients. The ...