2,798 research outputs found
A Coloring Algorithm for Disambiguating Graph and Map Drawings
Drawings of non-planar graphs always result in edge crossings. When there are
many edges crossing at small angles, it is often difficult to follow these
edges, because of the multiple visual paths resulted from the crossings that
slow down eye movements. In this paper we propose an algorithm that
disambiguates the edges with automatic selection of distinctive colors. Our
proposed algorithm computes a near optimal color assignment of a dual collision
graph, using a novel branch-and-bound procedure applied to a space
decomposition of the color gamut. We give examples demonstrating the
effectiveness of this approach in clarifying drawings of real world graphs and
maps
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
The edge chromatic number of outer-1-planar graphs
A graph is outer-1-planar if it can be drawn in the plane so that all
vertices are on the outer face and each edge is crossed at most once. In this
paper, we completely determine the edge chromatic number of outer 1-planar
graphs
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