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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, ${\displaystyle F}$ is a common graph if it "commonly" appears as a subgraph, in a sense that the total number of copies of ${\displaystyle F}$ in any graph ${\displaystyle G}$ and its complement ${\displaystyle {\overline {G}}}$ is a large fraction of all possible copies of ${\displaystyle F}$ on the same vertices. Intuitively, if ${\displaystyle G}$ contains few copies of ${\displaystyle F}$, then its complement ${\displaystyle {\overline {G}}}$ must contain lots of copies of ${\displaystyle F}$ 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

A graph ${\displaystyle F}$ is common if the inequality:

${\displaystyle t(F,W)+t(F,1-W)\geq 2^{-e(F)+1}}$

holds for any graphon ${\displaystyle W}$, where ${\displaystyle e(F)}$ is the number of edges of ${\displaystyle F}$ and ${\displaystyle t(F,W)}$ is the homomorphism density.

The inequality is tight because the lower bound is always reached when ${\displaystyle W}$ is the constant graphon ${\displaystyle W\equiv 1/2}$.

Interpretations of definition

For a graph ${\displaystyle G}$, we have ${\displaystyle t(F,G)=t(F,W_{G})}$ and ${\displaystyle t(F,{\overline {G}})=t(F,1-W_{G})}$ for the associated graphon ${\displaystyle W_{G}}$, since graphon associated to the complement ${\displaystyle {\overline {G}}}$ is ${\displaystyle W_{\overline {G}}=1-W_{G}}$. Hence, this formula provides us with the very informal intuition to take a close enough approximation, whatever that means, ${\displaystyle W}$ to ${\displaystyle W_{G}}$, and see ${\displaystyle t(F,W)}$ as roughly the fraction of labeled copies of graph ${\displaystyle F}$ in "approximate" graph ${\displaystyle G}$. Then, we can assume the quantity ${\displaystyle t(F,W)+t(F,1-W)}$ is roughly ${\displaystyle t(F,G)+t(F,{\overline {G}})}$ and interpret the latter as the combined number of copies of ${\displaystyle F}$ in ${\displaystyle G}$ and ${\displaystyle {\overline {G}}}$. Hence, we see that ${\displaystyle t(F,G)+t(F,{\overline {G}})\gtrsim 2^{-e(F)+1}}$ holds. This, in turn, means that common graph ${\displaystyle F}$ 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 ${\displaystyle 2^{-e(F)+1}}$ fraction of all possible copies of ${\displaystyle F}$ are monochromatic. Note that in a Erdős–Rényi random graph ${\displaystyle G=G(n,p)}$ with each edge drawn with probability ${\displaystyle p=1/2}$, each graph homomorphism from ${\displaystyle F}$ to ${\displaystyle G}$ have probability ${\displaystyle 2\cdot 2^{-e(F)}=2^{-e(F)+1}}$of being monochromatic. So, common graph ${\displaystyle F}$ is a graph where it attains its minimum number of appearance as a monochromatic subgraph of graph ${\displaystyle G}$ at the graph ${\displaystyle G=G(n,p)}$ with ${\displaystyle p=1/2}$

${\displaystyle p=1/2}$. The above definition using the generalized homomorphism density can be understood in this way.

Examples

• As stated above, all Sidorenko graphs are common graphs. Hence, any known Sidorenko graph is an example of a common graph, and, most notably, cycles of even length are common. 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 ${\displaystyle K_{3}}$ is one simple example of non-bipartite common graph.
• ${\displaystyle K_{4}^{-}}$, the graph obtained by removing an edge of the complete graph on 4 vertices ${\displaystyle K_{4}}$, is common.
• Non-example: It was believed for a time that all graphs are common. However, it turns out that ${\displaystyle K_{t}}$ is not common for ${\displaystyle t\geq 4}$. In particular, ${\displaystyle K_{4}}$ is not common even though ${\displaystyle K_{4}^{-}}$ is common.

Proofs

Sidorenko graphs are common

A graph ${\displaystyle F}$ is a Sidorenko graph if it satisfies ${\displaystyle t(F,W)\geq t(K_{2},W)^{e(F)}}$ for all graphons ${\displaystyle W}$.

In that case, ${\displaystyle t(F,1-W)\geq t(K_{2},1-W)^{e(F)}}$. Furthermore, ${\displaystyle t(K_{2},W)+t(K_{2},1-W)=1}$, which follows from the definition of homomorphism density. Combining this with Jensen's inequality for the function ${\displaystyle f(x)=x^{e(F)}}$:

${\displaystyle t(F,W)+t(F,1-W)\geq t(K_{2},W)^{e(F)}+t(K_{2},1-W)^{e(F)}\geq 2{\bigg (}{\frac {t(K_{2},W)+t(K_{2},1-W)}{2}}{\bigg )}^{e(F)}=2^{-e(F)+1}}$

Thus, the conditions for common graph is met.

The triangle graph is common

Expand the integral expression for ${\displaystyle t(K_{3},1-W)}$ and take into account the symmetry between the variables:

${\displaystyle \int _{^{3}}(1-W(x,y))(1-W(y,z))(1-W(z,x))dxdydz=1-3\int _{^{2}}W(x,y)+3\int _{^{3}}W(x,y)W(x,z)dxdydz-\int _{^{3}}W(x,y)W(y,z)W(z,x)dxdydz}$

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

${\displaystyle \int _{^{2}}W(x,y)dxdy=t(K_{2},W)}$

${\displaystyle \int {^{3}}W(x,y)W(x,z)dxdydz=t(K_{1,2},W)}$

${\displaystyle \int _{^{3}}W(x,y)W(y,z)W(z,x)dxdydz=t(K_{3},W)}$

where ${\displaystyle K_{1,2}}$ denotes the complete bipartite graph on ${\displaystyle 1}$ vertex on one part and ${\displaystyle 2}$ vertices on the other. It follows:

${\displaystyle t(K_{3},W)+t(K_{3},1-W)=1-3t(K_{2},W)+3t(K_{1,2},W)}$.

${\displaystyle t(K_{1,2},W)}$ can be related to ${\displaystyle t(K_{2},W)}$ thanks to the symmetry between the variables ${\displaystyle y}$ and ${\displaystyle z}$: $${\displaystyle {\begin{alignedat}{4}t(K_{1,2},W)&=\int _{^{3}}W(x,y)W(x,z)dxdydz&&\\&=\int _{x\in }{\bigg (}\int _{y\in }W(x,y){\bigg )}{\bigg (}\int _{z\in }W(x,z){\bigg )}&&\\&=\int _{x\in }{\bigg (}\int _{y\in }W(x,y){\bigg )}^{2}&&\\&\geq {\bigg (}\int _{x\in }\int _{y\in }W(x,y){\bigg )}^{2}=t(K_{2},W)^{2}\end{alignedat}}}$$

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

${\displaystyle t(K_{3},W)+t(K_{3},1-W)\geq 1-3t(K_{2},W)+3t(K_{2},W)^{2}=1/4+3{\big (}t(K_{2},W)-1/2{\big )}^{2}\geq 1/4}$.

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

See also

• Sidorenko's conjecture

References

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.

• Graph families
• Extremal graph theory