140 research outputs found
Coloring Complexes and Combinatorial Hopf Monoids
We generalize the notion of coloring complex of a graph to linearized
combinatorial Hopf monoids. These are a generalization of the notion of
coloring complex of a graph. We determine when a combinatorial Hopf monoid has
such a construction, and discover some inequalities that are satisfied by the
quasisymmetric function invariants associated to the combinatorial Hopf monoid.
We show that the collection of all such coloring complexes forms a
combinatorial Hopf monoid, which is the terminal object in the category of
combinatorial Hopf monoids with convex characters. We also study several
examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Network Coding in a Multicast Switch
We consider the problem of serving multicast flows in a crossbar switch. We
show that linear network coding across packets of a flow can sustain traffic
patterns that cannot be served if network coding were not allowed. Thus,
network coding leads to a larger rate region in a multicast crossbar switch. We
demonstrate a traffic pattern which requires a switch speedup if coding is not
allowed, whereas, with coding the speedup requirement is eliminated completely.
In addition to throughput benefits, coding simplifies the characterization of
the rate region. We give a graph-theoretic characterization of the rate region
with fanout splitting and intra-flow coding, in terms of the stable set
polytope of the 'enhanced conflict graph' of the traffic pattern. Such a
formulation is not known in the case of fanout splitting without coding. We
show that computing the offline schedule (i.e. using prior knowledge of the
flow arrival rates) can be reduced to certain graph coloring problems. Finally,
we propose online algorithms (i.e. using only the current queue occupancy
information) for multicast scheduling based on our graph-theoretic formulation.
In particular, we show that a maximum weighted stable set algorithm stabilizes
the queues for all rates within the rate region.Comment: 9 pages, submitted to IEEE INFOCOM 200
New bounds for the max--cut and chromatic number of a graph
We consider several semidefinite programming relaxations for the max--cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max--cut when that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max--cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme
Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory
International audienceWe investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions
Connection between Graphs' Chromatic and Ehrhart Polynomials
Graph Theory is a discipline of mathematics with numerous outstanding issues and applications in a variety of sectors of mathematics and science. The chromatic polynomial is a type of polynomial that has useful and attractive qualities. Ehrhart's polynomials and chromatic analysis are two essential techniques for graph analysis. They both provide insight into the graph's structure but in different ways. The relationship between chromatic and Ehrhart polynomials is an area of active research that has implications for graph theory, combinatorial, and other fields. By understanding the relationship between these two polynomials, one can better understand the structure of graphs and how they interact with each other. This can help us to solve complex problems in our lives more efficiently and effectively. This work gives the relationship between these two essential polynomials and the proof of theorems and an application related to these works was discussed, which is the model Physical Cell ID (PCID)
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