4,617 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
Unknotting numbers and triple point cancelling numbers of torus-covering knots
It is known that any surface knot can be transformed to an unknotted surface
knot or a surface knot which has a diagram with no triple points by a finite
number of 1-handle additions. The minimum number of such 1-handles is called
the unknotting number or the triple point cancelling number, respectively. In
this paper, we give upper bounds and lower bounds of unknotting numbers and
triple point cancelling numbers of torus-covering knots, which are surface
knots in the form of coverings over the standard torus . Upper bounds are
given by using -charts on presenting torus-covering knots, and lower
bounds are given by using quandle colorings and quandle cocycle invariants.Comment: 26 pages, 14 figures, added Corollary 1.7, to appear in J. Knot
Theory Ramification
The Coloring Ideal and Coloring Complex of a Graph
Let be a simple graph on vertices. We define a monomial ideal in
the Stanley-Reisner ring of the order complex of the Boolean algebra on
atoms. The monomials in are in one-to-one correspondence with the proper
colorings of . In particular, the Hilbert polynomial of equals the
chromatic polynomial of .
The ideal is generated by square-free monomials, so is the
Stanley-Reisner ring of a simplicial complex . The -vector of is a
certain transformation of the tail of the chromatic polynomial
of . The combinatorial structure of the complex is described
explicitly and it is shown that the Euler characteristic of equals the
number of acyclic orientations of .Comment: 13 pages, 3 figure
A blow-up construction and graph coloring
Given a graph G (or more generally a matroid embedded in a projective space),
we construct a sequence of varieties whose geometry encodes combinatorial
information about G. For example, the chromatic polynomial of G (giving at each
m>0 the number of colorings of G with m colors, such that no adjacent vertices
are assigned the same color) can be computed as an intersection product between
certain classes on these varieties, and other information such as Crapo's
invariant find a very natural geometric counterpart. The note presents this
construction, and gives `geometric' proofs of a number of standard
combinatorial results on the chromatic polynomial.Comment: 22 pages, amstex 2.
On the fine-grained complexity of rainbow coloring
The Rainbow k-Coloring problem asks whether the edges of a given graph can be
colored in colors so that every pair of vertices is connected by a rainbow
path, i.e., a path with all edges of different colors. Our main result states
that for any , there is no algorithm for Rainbow k-Coloring running in
time , unless ETH fails.
Motivated by this negative result we consider two parameterized variants of
the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et
al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set of
pairs of vertices and we ask if there is a coloring in which all the pairs in
are connected by rainbow paths. We show that Subset Rainbow k-Coloring is
FPT when parameterized by . We also study Maximum Rainbow k-Coloring
problem, where we are additionally given an integer and we ask if there is
a coloring in which at least anti-edges are connected by rainbow paths. We
show that the problem is FPT when parameterized by and has a kernel of size
for every (thus proving that the problem is FPT), extending the
result of Ananth et al. [FSTTCS 2011]
Directed paths with few or many colors in colored directed graphs
Given a graph and a coloring of , not necessarily a proper coloring of either the arcs or the vertices of , we consider the complexity of finding a path of from a given vertex to another given vertex with as few different colors as possible, and of finding one with as many different colors as possible. We show that the first problem is polynomial-time solvable, and that the second problem is NP-hard. \u
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