5,009 research outputs found
Coloring non-crossing strings
For a family of geometric objects in the plane
, define as the least
integer such that the elements of can be colored with
colors, in such a way that any two intersecting objects have distinct
colors. When is a set of pseudo-disks that may only intersect on
their boundaries, and such that any point of the plane is contained in at most
pseudo-disks, it can be proven that
since the problem is equivalent to cyclic coloring of plane graphs. In this
paper, we study the same problem when pseudo-disks are replaced by a family
of pseudo-segments (a.k.a. strings) that do not cross. In other
words, any two strings of are only allowed to "touch" each other.
Such a family is said to be -touching if no point of the plane is contained
in more than elements of . We give bounds on
as a function of , and in particular we show that
-touching segments can be colored with colors. This partially answers
a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems
of strings.Comment: 19 pages. A preliminary version of this work appeared in the
proceedings of EuroComb'09 under the title "Coloring a set of touching
strings
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 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
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