A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem

A note on recognizing an old friend in a new place:list coloring and the zero-temperature Potts model

Here we observe that list coloring in graph theory coincides with the zero-temperature antiferromagnetic Potts model with an external field. We give a list coloring polynomial that equals the partition function in this case. This is analogous to the well-known connection between the chromatic polynomial and the zero-temperature, zero-field, antiferromagnetic Potts model. The subsequent cross fertilization yields immediate results for the Potts model and suggests new research directions in list coloring

The Las Vergnas Polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas' results on Eulerian circuits from graphs in surfaces of low genus to surfaces of arbitrary genus

DNA origami and the complexity of Eulerian circuits with turning costs

Building a structure using self-assembly of DNA molecules by origami folding requires finding a route for the scaffolding strand through the desired structure. When the target structure is a 1-complex (or the geometric realization of a graph), an optimal route corresponds to an Eulerian circuit through the graph with minimum turning cost. By showing that it leads to a solution to the 3-SAT problem, we prove that the general problem of finding an optimal route for a scaffolding strand for such structures is NP-hard. We then show that the problem may readily be transformed into a Traveling Salesman Problem (TSP), so that machinery that has been developed for the TSP may be applied to find optimal routes for the scaffolding strand in a DNA origami self-assembly process. We give results for a few special cases, showing for example that the problem remains intractable for graphs with maximum degree 8, but is polynomial time for 4-regular plane graphs if the circuit is restricted to following faces. We conclude with some implications of these results for related problems, such as biomolecular computing and mill routing problems

Method specific calibration corrects for DNA extraction method effects on relative telomere length measurements by quantitative PCR

Telomere length (TL) is increasingly being used as a biomarker in epidemiological, biomedical and ecological studies. A wide range of DNA extraction techniques have been used in telomere experiments and recent quantitative PCR (qPCR) based studies suggest that the choice of DNA extraction method may influence average relative TL (RTL) measurements. Such extraction method effects may limit the use of historically collected DNA samples extracted with different methods. However, if extraction method effects are systematic an extraction method specific (MS) calibrator might be able to correct for them, because systematic effects would influence the calibrator sample in the same way as all other samples. In the present study we tested whether leukocyte RTL in blood samples from Holstein Friesian cattle and Soay sheep measured by qPCR was influenced by DNA extraction method and whether MS calibration could account for any observed differences. We compared two silica membrane-based DNA extraction kits and a salting out method. All extraction methods were optimized to yield enough high quality DNA for TL measurement. In both species we found that silica membrane-based DNA extraction methods produced shorter RTL measurements than the non-membrane-based method when calibrated against an identical calibrator. However, these differences were not statistically detectable when a MS calibrator was used to calculate RTL. This approach produced RTL measurements that were highly correlated across extraction methods (r > 0.76) and had coefficients of variation lower than 10% across plates of identical samples extracted by different methods. Our results are consistent with previous findings that popular membrane-based DNA extraction methods may lead to shorter RTL measurements than non-membrane-based methods. However, we also demonstrate that these differences can be accounted for by using an extraction method-specific calibrator, offering researchers a simple means of accounting for differences in RTL measurements from samples extracted by different DNA extraction methods within a study