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.

Definition
A graph $$F$$ is common if the inequality:

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

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

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

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, it turns out that $$K_{t}$$ is not common for $$t \ge 4$$. In particular, $$K_4$$ is not common even though $$K_{4} ^{-}$$ is common.

Sidorenko graphs are common
A graph $$F$$ is a Sidorenko graph if it satisfies $$t(F, W) \ge t(K_2, W)^{e(F)}$$ for all graphons $$W$$.

In that case, $$t(F, 1 - W) \ge t(K_2, 1 - W)^{e(F)}$$. Furthermore, $$t(K_2, W) + t(K_2, 1 - W) = 1 $$, which follows from the definition of homomorphism density. Combining this with Jensen's inequality for the function $$f(x) = x^{e(F)}$$:

$$t(F, W) + t(F, 1 - W) \ge t(K_2, W)^{e(F)} + t(K_2, 1 - W)^{e(F)} \ge 2 \bigg( \frac{t(K_2, W) + t(K_2, 1 - W)}{2} \bigg)^{e(F)} = 2^{-e(F) + 1}$$

Thus, the conditions for common graph is met.

The triangle graph is common
Expand the integral expression for $$t(K_3, 1 - W)$$ and take into account the symmetry between the variables:

$$\int_{[0, 1]^3} (1 - W(x, y))(1 - W(y, z))(1 - W(z, x)) dx dy dz = 1 - 3 \int_{[0, 1]^2} W(x, y) + 3 \int_{[0, 1]^3} W(x, y) W(x, z) dx dy dz - \int_{[0, 1]^3} W(x, y) W(y, z) W(z, x) dx dy dz$$

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


 * $$\int_{[0, 1]^2} W(x, y) dx dy = t(K_2, W) $$
 * $$\int{[0, 1]^3} W(x, y) W(x, z) dx dy dz = t(K_{1, 2}, W) $$
 * $$\int_{[0, 1]^3} W(x, y) W(y, z) W(z, x) dx dy dz = t(K_3, W)$$

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


 * $$t(K_3, W) + t(K_3, 1 - W) = 1 - 3 t(K_2, W) + 3 t(K_{1, 2}, W) $$.

$$t(K_{1, 2}, W)$$ can be related to $$t(K_2, W)$$ thanks to the symmetry between the variables $$y $$ and $$z$$: $$\begin{alignat}{4} t(K_{1, 2}, W) &= \int_{[0, 1]^3} W(x, y) W(x, z) dx dy dz && \\ &= \int_{x \in [0, 1]} \bigg( \int_{y \in [0, 1]} W(x, y) \bigg) \bigg( \int_{z \in [0, 1]} W(x, z) \bigg) && \\ &= \int_{x \in [0, 1]} \bigg( \int_{y \in [0, 1]} W(x, y) \bigg)^2 && \\ &\ge \bigg( \int_{x \in [0, 1]} \int_{y \in [0, 1]} W(x, y) \bigg)^2 = t(K_2, W)^2 \end{alignat}$$

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

$$t(K_3, W) + t(K_3, 1 - W) \ge 1 - 3 t(K_2, W) + 3 t(K_{2}, W)^2 = 1/4 + 3 \big( t(K_2, W) - 1/2 \big)^2 \ge 1/4$$.

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