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Common graph

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In graph theory, an area of mathematics, common graphs belong to a branch of extremal graph theory concerning inequalities in homomorphism densities. Roughly speaking, is a common graph if it "commonly" appears as a subgraph, in a sense that the total number of copies of in any graph and its complement is a large fraction of all possible copies of on the same vertices. Intuitively, if contains few copies of , then its complement must contain lots of copies of in order to compensate for it.

Common graphs are closely related to other graph notions dealing with homomorphism density inequalities. For example, common graphs are a more general case of Sidorenko graphs.

Definition

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A graph is common if the inequality:

holds for any graphon , where is the number of edges of and is the homomorphism density.[1]

The inequality is tight because the lower bound is always reached when is the constant graphon .

Interpretations of definition

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For a graph , we have and for the associated graphon , since graphon associated to the complement is . Hence, this formula provides us with the very informal intuition to take a close enough approximation, whatever that means,[2] to , and see as roughly the fraction of labeled copies of graph in "approximate" graph . Then, we can assume the quantity is roughly and interpret the latter as the combined number of copies of in and . Hence, we see that holds. This, in turn, means that common graph commonly appears as subgraph.

In other words, if we think of edges and non-edges as 2-coloring of edges of complete graph on the same vertices, then at least fraction of all possible copies of are monochromatic. Note that in a Erdős–Rényi random graph with each edge drawn with probability , each graph homomorphism from to have probability of being monochromatic. So, common graph is a graph where it attains its minimum number of appearance as a monochromatic subgraph of graph at the graph with

. The above definition using the generalized homomorphism density can be understood in this way.

Examples

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  • As stated above, all Sidorenko graphs are common graphs.[3] Hence, any known Sidorenko graph is an example of a common graph, and, most notably, cycles of even length are common.[4] However, these are limited examples since all Sidorenko graphs are bipartite graphs while there exist non-bipartite common graphs, as demonstrated below.
  • The triangle graph is one simple example of non-bipartite common graph.[5]
  • , the graph obtained by removing an edge of the complete graph on 4 vertices , is common.[6]
  • Non-example: It was believed for a time that all graphs are common. However, it turns out that is not common for .[7] In particular, is not common even though is common.

Proofs

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Sidorenko graphs are common

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A graph is a Sidorenko graph if it satisfies for all graphons .

In that case, . Furthermore, , which follows from the definition of homomorphism density. Combining this with Jensen's inequality for the function :

Thus, the conditions for common graph is met.[8]

The triangle graph is common

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Expand the integral expression for and take into account the symmetry between the variables:

Each term in the expression can be written in terms of homomorphism densities of smaller graphs. By the definition of homomorphism densities:

where denotes the complete bipartite graph on vertex on one part and vertices on the other. It follows:

.

can be related to thanks to the symmetry between the variables and :

where the last step follows from the integral Cauchy–Schwarz inequality. Finally:

.

This proof can be obtained from taking the continuous analog of Theorem 1 in "On Sets Of Acquaintances And Strangers At Any Party"[9]

See also

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References

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  1. ^ Large Networks and Graph Limits. American Mathematical Society. p. 297. Retrieved 2022-01-13.
  2. ^ Borgs, C.; Chayes, J. T.; Lovász, L.; Sós, V. T.; Vesztergombi, K. (2008-12-20). "Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing". Advances in Mathematics. 219 (6): 1801–1851. arXiv:math/0702004. doi:10.1016/j.aim.2008.07.008. ISSN 0001-8708. S2CID 5974912.
  3. ^ Large Networks and Graph Limits. American Mathematical Society. p. 297. Retrieved 2022-01-13.
  4. ^ Sidorenko, A. F. (1992). "Inequalities for functionals generated by bipartite graphs". Discrete Mathematics and Applications. 2 (5). doi:10.1515/dma.1992.2.5.489. ISSN 0924-9265. S2CID 117471984.
  5. ^ Large Networks and Graph Limits. American Mathematical Society. p. 299. Retrieved 2022-01-13.
  6. ^ Large Networks and Graph Limits. American Mathematical Society. p. 298. Retrieved 2022-01-13.
  7. ^ Thomason, Andrew (1989). "A Disproof of a Conjecture of Erdős in Ramsey Theory". Journal of the London Mathematical Society. s2-39 (2): 246–255. doi:10.1112/jlms/s2-39.2.246. ISSN 1469-7750.
  8. ^ Lovász, László (2012). Large Networks and Graph Limits. United States: American Mathematical Society Colloquium publications. pp. 297–298. ISBN 978-0821890851.
  9. ^ Goodman, A. W. (1959). "On Sets of Acquaintances and Strangers at any Party". The American Mathematical Monthly. 66 (9): 778–783. doi:10.2307/2310464. ISSN 0002-9890. JSTOR 2310464.