869 research outputs found

    Universal targets for homomorphisms of edge-colored graphs

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    A kk-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,…,k1,\ldots,k. A function from the vertex set of one kk-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 F\mathcal{F} of graphs, a kk-edge-colored graph H\mathbb{H} (not necessarily with the underlying graph in F\mathcal{F}) is kk-universal for F\mathcal{F} when any kk-edge-colored graph with the underlying graph in F\mathcal{F} admits a homomorphism to H\mathbb{H}. We characterize graph classes that admit kk-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph. For a nonempty graph GG, the density of GG is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of GG. For a nonempty class F\mathcal{F} of graphs, D(F)D(\mathcal{F}) denotes the density of F\mathcal{F}, that is the supremum of densities of graphs in F\mathcal{F}. The main results are the following. The class F\mathcal{F} admits kk-universal graphs for k≥2k\geq2 if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in F\mathcal{F}. For any such class, there exists a constant cc, such that for any k≥2k \geq 2, the size of the smallest kk-universal graph is between kD(F)k^{D(\mathcal{F})} and ck⌈D(F)⌉ck^{\lceil D(\mathcal{F})\rceil}. 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 kk-universal graph is between k3+3k^3+3 and 5k45k^4. Our results yield that there exists a constant cc such that for all kk, this size is bounded from above by ck3ck^3

    Modified 6j-symbols and 3-manifold invariants

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    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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