4,617 research outputs found

    On signed diagonal flip sequences

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    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

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    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 TT. Upper bounds are given by using mm-charts on TT 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

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    Let GG be a simple graph on dd vertices. We define a monomial ideal KK in the Stanley-Reisner ring AA of the order complex of the Boolean algebra on dd atoms. The monomials in KK are in one-to-one correspondence with the proper colorings of GG. In particular, the Hilbert polynomial of KK equals the chromatic polynomial of GG. The ideal KK is generated by square-free monomials, so A/KA/K is the Stanley-Reisner ring of a simplicial complex CC. The hh-vector of CC is a certain transformation of the tail T(n)=ndk(n)T(n)= n^d-k(n) of the chromatic polynomial kk of GG. The combinatorial structure of the complex CC is described explicitly and it is shown that the Euler characteristic of CC equals the number of acyclic orientations of GG.Comment: 13 pages, 3 figure

    A blow-up construction and graph coloring

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    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

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    The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in kk 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 k2k\ge 2, there is no algorithm for Rainbow k-Coloring running in time 2o(n3/2)2^{o(n^{3/2})}, 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 SS of pairs of vertices and we ask if there is a coloring in which all the pairs in SS are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by S|S|. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer qq and we ask if there is a coloring in which at least qq anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by qq and has a kernel of size O(q)O(q) for every k2k\ge 2 (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

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    Given a graph D=(V(D),A(D))D=(V(D),A(D)) and a coloring of DD, not necessarily a proper coloring of either the arcs or the vertices of DD, we consider the complexity of finding a path of DD from a given vertex ss to another given vertex tt 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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