Delocalization of Uniform Graph Homomorphisms from $${\mathbb {Z}}^2$$ to $${\mathbb {Z}}$$

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

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Proof of an Intersection Theorem via Graph Homomorphisms

The Electronic Journal of Combinatorics ◽

10.37236/1144 ◽

2006 ◽

Vol 13 (1) ◽

Cited By ~ 2

Author(s):

Irit Dinur ◽

Ehud Friedgut

Keyword(s):

Product Measure ◽

Intersection Theorem ◽

Intersecting Family ◽

Graph Homomorphisms

Let $0 \leq p \leq 1/2 $ and let $\{0,1\}^n$ be endowed with the product measure $\mu_p$ defined by $\mu_p(x)=p^{|x|}(1-p)^{n-|x|}$, where $|x|=\sum x_i$. Let $I \subseteq \{0,1\}^n$ be an intersecting family, i.e. for every $x, y \in I$ there exists a coordinate $1 \leq i \leq n$ such that $x_i=y_i=1$. Then $\mu_p(I) \leq p.$ Our proof uses measure preserving homomorphisms between graphs.

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Infinite limits and folding

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3410 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Anthony Bonato ◽

Jeannette Janssen

Keyword(s):

Biological Networks ◽

Infinite Graphs ◽

Isomorphism Type ◽

Connected Component ◽

Induced Subgraph ◽

Pursuit Games ◽

Graph Homomorphisms ◽

International Audience

International audience We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.

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