9,827 research outputs found
Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs
Given a multigraph , the {\em edge-coloring problem} (ECP) is to
color the edges of with the minimum number of colors so that no two
adjacent edges have the same color. This problem can be naturally formulated as
an integer program, and its linear programming relaxation is called the {\em
fractional edge-coloring problem} (FECP). In the literature, the optimal value
of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional
chromatic index}) of , denoted by (resp. ). Let
be the maximum degree of and let where is the set of all edges of with
both ends in . Clearly, is
a lower bound for . As shown by Seymour, . In the 1970s Goldberg and Seymour independently conjectured
that . Over the
past four decades this conjecture, a cornerstone in modern edge-coloring, has
been a subject of extensive research, and has stimulated a significant body of
work. In this paper we present a proof of this conjecture. Our result implies
that, first, there are only two possible values for , so an analogue
to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in
graph theory, holds for multigraphs; second, although it is -hard in
general to determine , we can approximate it within one of its true
value, and find it exactly in polynomial time when ;
third, every multigraph satisfies , so FECP has a
fascinating integer rounding property
Topological quantum phase transition in an S=2 spin chain
We construct a model Hamiltonian for S = 2 spin chain, where a variable
parameter is introduced. The edge spin is S = 1 for , and
S = 3/2 for . Due to the topological distinction of the edge
states, these two phases must be separated by one or several topological
quantum phase transitions. We investigate the nature of the quantum phase
transition by DMRG calculation, and propose a phase diagram for this model.Comment: 5 pages, 4 figure
An efficient algorithm for finding maximum cycle packings in reducible flow graphs
Reducible flow graphs occur naturally in connection with flow-charts of computer programs and are used extensively for code optimization and global data flow analysis. In this paper we present an O(n2m log (n 2/m)) algorithm for finding a maximum cycle packing in any weighted reducible flow graph with n vertices and m arcs. Β© Springer-Verlag 2004.postprin
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