4,485 research outputs found
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
On Cyclic Edge-Connectivity of Fullerenes
A graph is said to be cyclic -edge-connected, if at least edges must
be removed to disconnect it into two components, each containing a cycle. Such
a set of edges is called a cyclic--edge cutset and it is called a
trivial cyclic--edge cutset if at least one of the resulting two components
induces a single -cycle.
It is known that fullerenes, that is, 3-connected cubic planar graphs all of
whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this
article it is shown that a fullerene containing a nontrivial cyclic-5-edge
cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces
whose neighboring faces are also pentagonal. Moreover, it is shown that has
a Hamilton cycle, and as a consequence at least perfect matchings, where is the order of .Comment: 11 pages, 9 figure
A Self-Linking Invariant of Virtual Knots
In this paper we introduce a new invariant of virtual knots and links that is
non-trivial for infinitely many virtuals, but is trivial on classical knots and
links. The invariant is initially be expressed in terms of a relative of the
bracket polynomial and then extracted from this polynomial in terms of its
exponents, particularly for the case of knots. This analog of the bracket
polynomial will be denoted {K} (with curly brackets) and called the binary
bracket polynomial. The key to the combinatorics of the invariant is an
interpretation of the state sum in terms of 2-colorings of the associated
diagrams. For virtual knots, the new invariant, J(K), is a restriction of the
writhe to the odd crossings of the diagram (A crossing is odd if it links an
odd number of crossings in the Gauss code of the knot. The set of odd crossings
is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies
that K is non-trivial, non-classical and inequivalent to its planar mirror
image. The paper also condsiders generalizations of the two-fold coloring of
the states of the binary bracket to cases of three and more colors.
Relationships with graph coloring and the Four Color Theorem are discussed.Comment: 36 pages, 22 figures, LaTeX documen
Disjoint compatibility graph of non-crossing matchings of points in convex position
Let be a set of labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
. Two such matchings, and , are disjoint compatible if they do
not have common edges, and no edge of crosses an edge of . Denote by
the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each , the connected components of
form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 figure
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Injectivity radius of representations of triangle groups and planar width of regular hypermaps
We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width
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