Representations of graphs modulo n

Abstract

A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

Identification of a hydrogen bond in the Phe M197→Tyr mutant reaction center of the photosynthetic purple bacterium Rhodobacter sphaeroides by X‐ray crystallography and FTIR spectroscopy1

“…Geometric parameters of Engh and Huber [24] for the polypeptide chain and parameters of Treutlein et al [25], Lancaster and Michel [26], and U. Ermler (unpublished data) for the cofactors were applied in the crystallographic re¢nement. Manual rebuilding of the structure model was performed with the graphics program O [27], and the structure ¢gures were produced with the graphics program SETOR [28].…”

Mutational analysis of the Mycobacterium tuberculosis Rv1625c adenylyl cyclase: residues that confer nucleotide specificity contribute to dimerization

“…The predicted salt bridge between these residues and an interaction between the aromatic groups of the side chains of Phe-363 residues from the two subunits are represented by dotted spheres. The ¢gure was generated using SETOR [20].…”

Crystal structure of neuroserpin: a neuronal serpin involved in a conformational disease

“…The loss of the hydrogen bond between Asn186 and His334 is compensated for by the formation of a hydrogen bond with the OH side chain of Ser351. Prepared with Setor [31]. Fig.…”

“…To fully insert the RCL, Arg56 must lose its hydrogen bonds and shift into another position, that is also slowing down the rate of the full insertion. Prepared with Setor [31]. 6…”