870 research outputs found
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Modified 6j-symbols and 3-manifold invariants
37 pages, 16 figuresInternational audienceWe show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects
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