778 research outputs found
Hexagonal Tilings and Locally C6 Graphs
We give a complete classification of hexagonal tilings and locally C6 graphs,
by showing that each of them has a natural embedding in the torus or in the
Klein bottle. We also show that locally grid graphs are minors of hexagonal
tilings (and by duality of locally C6 graphs) by contraction of a perfect
matching and deletion of the resulting parallel edges, in a form suitable for
the study of their Tutte uniqueness.Comment: 14 figure
Hexagonal Tilings: Tutte Uniqueness
We develop the necessary machinery in order to prove that hexagonal tilings
are uniquely determined by their Tutte polynomial, showing as an example how to
apply this technique to the toroidal hexagonal tiling.Comment: 12 figure
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Chromaticity of Certain Bipartite Graphs
Since the introduction of the concepts of chromatically unique graphs and chromatically
equivalent graphs, numerous families of such graphs have been obtained.
The purpose of this thesis is to continue with the search of families of
chromatically unique bipartite graphs.
In Chapters 1 and 2, we define the concept of graph colouring, the associated
chromatic polynomial and some properties of a chromatic polynomial. We also
give some necessary conditions for graphs that are chromatically unique or chromatically
equivalent. We end this chapter by stating some known results on the
chromaticity of bipartite graphs, denoted as K(p,q)
New Bounds for the Dichromatic Number of a Digraph
The chromatic number of a graph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that each
color class is an independent set (a set of pairwise non-adjacent vertices).
The dichromatic number of a digraph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that
each color class is acyclic.
In 1976, Bondy proved that the chromatic number of a digraph is at most
its circumference, the length of a longest cycle.
Given a digraph , we will construct three different graphs whose chromatic
numbers bound .
Moreover, we prove: i) for integers , and with and for each , that if all
cycles in have length modulo for some ,
then ; ii) if has girth and there are integers
and , with such that contains no cycle of length
modulo for each , then ;
iii) if has girth , the length of a shortest cycle, and circumference
, then , which improves,
substantially, the bound proposed by Bondy. Our results show that if we have
more information about the lengths of cycles in a digraph, then we can improve
the bounds for the dichromatic number known until now.Comment: 14 page
Graphs determined by polynomial invariants
AbstractMany polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results
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