12 research outputs found
On signed diagonal flip sequences
Eliahou \cite{2} and Kryuchkov \cite{9} conjectured a proposition that
Gravier and Payan \cite{4} proved to be equivalent to the Four Color Theorem.
It states that any triangulation of a polygon can be transformed into another
triangulation of the same polygon by a sequence of signed diagonal flips. It is
well known that any pair of polygonal triangulations are connected by a
sequence of (non-signed) diagonal flips. In this paper we give a sufficient and
necessary condition for a diagonal flip sequence to be a signed diagonal flip
sequence.Comment: 11 pages, 24 figures, to appear in European Journal of Combinatoric
Three-Colorings of Cubic Graphs and Tensor Operators
Penrose's work \cite{8} established a connection between the edge 3-colorings
of cubic planar graphs and tensor algebras. We exploit this point of view in
order to get algebraic representations of the category of cubic graphs with
free ends.Comment: 11 pages, many figures and diagram