38 research outputs found
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
A coarse Tutte polynomial for hypermaps
We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction relations where the base case consists of isolated vertices. Our Tutte polynomial extends the classical Tutte polynomial of a graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon graph polynomial). We examine relations between our polynomial and other hypermap polynomials. We give hypermap duality and partial duality identities for our polynomial, as well as some evaluations
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