Common graph

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.

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.^[1]

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

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,^[2] ${\displaystyle W}$ 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.

• 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 ${\displaystyle K_{3}}$ is one simple example of non-bipartite common graph.^[5]
• ${\displaystyle K_{4}^{-}}$, the graph obtained by removing an edge of the complete graph on 4 vertices ${\displaystyle K_{4}}$, is common.^[6]
• 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 ${\displaystyle t\geq 4}$.^[7] In particular, ${\displaystyle K_{4}}$ is not common even though ${\displaystyle K_{4}^{-}}$ is 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.^[8]

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

${\displaystyle \int _{[0,1]^{3}}(1-W(x,y))(1-W(y,z))(1-W(z,x))dxdydz=1-3\int _{[0,1]^{2}}W(x,y)+3\int _{[0,1]^{3}}W(x,y)W(x,z)dxdydz-\int _{[0,1]^{3}}W(x,y)W(y,z)W(z,x)dxdydz}$

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

${\displaystyle \int _{[0,1]^{2}}W(x,y)dxdy=t(K_{2},W)}$

${\displaystyle \int {[0,1]^{3}}W(x,y)W(x,z)dxdydz=t(K_{1,2},W)}$

${\displaystyle \int _{[0,1]^{3}}W(x,y)W(y,z)W(z,x)dxdydz=t(K_{3},W)}$

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 ${\displaystyle z}$: $${\displaystyle {\begin{alignedat}{4}t(K_{1,2},W)&=\int _{[0,1]^{3}}W(x,y)W(x,z)dxdydz&&\\&=\int _{x\in [0,1]}{\bigg (}\int _{y\in [0,1]}W(x,y){\bigg )}{\bigg (}\int _{z\in [0,1]}W(x,z){\bigg )}&&\\&=\int _{x\in [0,1]}{\bigg (}\int _{y\in [0,1]}W(x,y){\bigg )}^{2}&&\\&\geq {\bigg (}\int _{x\in [0,1]}\int _{y\in [0,1]}W(x,y){\bigg )}^{2}=t(K_{2},W)^{2}\end{alignedat}}}$$

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"^[9]

• Sidorenko's conjecture