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

Definition

Formally, a common graph is a graph $F$ such that the inequality:

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

holds for any graphon $W$, where $e(F)$ is the number of edges of $F$ and $t(F,W)$ is the homomorphism density (see the book "Large Networks and Graph Limits". and a survey "Very Large Graphs" , both by László Lovász, for introduction to the theory of graph limits). Here, note that the inequality attains the lower bound when $W$ is the constant graphon $W\equiv 1/2$. So, the inequality is tight.       

Interpretations of definition

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

$p=1/2$. 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 $K_{3}$ is one simple example of non-bipartite common graph.
• $K_{4}^{-}$, the graph obtained by removing an edge of the complete graph on 4 vertices $K_{4}$, is common.
• Non-example: It was believed for a time that all graphs are common. However, shockingly, it turns out that $K_{t}$ is not common for $t\geq 4$, as proved by Thomason in 1989. In particular, $K_{4}$ is not common even though $K_{4}^{-}$ is common.

Proofs

In this section, we will prove some of the above examples.

Sidorenko graphs are common

Recall that a Sidorenko graph $F$ is a graph satisfying $t(F,W)\geq t(K_{2},W)^{e(F)}$ for all graphons $W$. Hence, we should also have $t(F,1-W)\geq t(K_{2},1-W)^{e(F)}$. Now, observe that $t(K_{2},W)+t(K_{2},1-W)=1$, which follows from the definition of homomorphism density. Combining this with Jensen's inequality for the function $f(x)=x^{e(F)}$, we can see that

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

Thus, the conditions for common graph is met. This proof, along with proof of $K_{4}^{-}$ being common graph, can be seen from page 297-298 of the aforementioned book of László Lovász.

The triangle graph is common

Here, we will expand the integral expression for $t(K_{3},1-W)$ and take into account the symmetry between the variables:

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

Now, observe that each term in the expression can be written in terms of homomorphism densities of smaller graphs. Indeed, by the definition of homomorphism densities, we have:

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

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

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

(Note that $K_{1,2}$ denotes the complete bipartite graph on $1$ vertex on one part and $2$ vertices on the other.) Hence, we get:

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

Now, in order to relate $t(K_{1,2},W)$ to $t(K_{2},W)$, note that we can exploit the symmetry between the variables $y$ and $z$ to write:$${\begin{alignedat}{4}t(K_{1,2},W)&=\int _{[0,1]^{3}}W(x,y)W(x,z)dxdydz&&\\&=\int _{x\in [0,1]}{\bigg (}\int _{y\in [0,1]}W(x,y){\bigg )}{\bigg (}\int _{z\in [0,1]}W(x,z){\bigg )}&&\\&=\int _{x\in [0,1]}{\bigg (}\int _{y\in [0,1]}W(x,y){\bigg )}^{2}&&\\&\geq {\bigg (}\int _{x\in [0,1]}\int _{y\in [0,1]}W(x,y){\bigg )}^{2}=t(K_{2},W)^{2}\end{alignedat}}$$where we used the integral Cauchy–Schwarz inequality in the last step. Finally, our desired result follows from the above inequality:

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

This above proof can be obtained from taking continuous analog of Theorem 1 in Goodman 1959 paper, "On Sets Of Acquaintances And Strangers At Any Party"

See also

• Sidorenko's conjecture