Common graph: Difference between revisions

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Formally, a common graph is a graph <math>F</math> such that the inequality:

<math>t(F, W) + t(F, 1 - W) \ge 2^{-e(F)+1}</math>

holds for any [[graphon]] <math>W</math>, where <math>e(F)</math> is the number of edges of <math>F</math> and <math>t(F, W)</math> is the [[homomorphism density]] (see the book "''Large Networks and Graph Limits''"<ref>{{Cite web|title=Large Networks and Graph Limits|url=https://bookstore.ams.org/coll-60/|access-date=2022-01-13|website=bookstore.ams.org}}</ref> and a survey "''Very Large Graphs''"<ref>{{Cite journal|last=Lovasz|first=Laszlo|date=2009-02-01|title=Very large graphs|url=http://arxiv.org/abs/0902.0132|journal=arXiv:0902.0132 [math]}}</ref> , both by [[László Lovász]], for introduction to the theory of [[graph limits]]). Here, note that the inequality attains the lower bound when <math>W</math> is the constant graphon <math>W \equiv 1/2</math>. So, the inequality is tight.       

== Interpretations of definition ==

For a graph <math>G</math>, we have <math>t(F, G) = t(F, W_{G}) </math> and <math>t(F, \overline{G})=t(F, 1 - W_G)</math> for the [[Graphon#Analytic Formulation|associated graphon]] <math>W_G</math>, since graphon associated to the complement <math>\overline{G}</math> is <math>W_{\overline{G}}=1 - W_G</math>. Hence, this formula provides us with the very informal intuition to take a close enough approximation, whatever that means,<ref>{{Cite journal|last=Borgs|first=C.|last2=Chayes|first2=J. T.|last3=Lovász|first3=L.|last4=Sós|first4=V. T.|last5=Vesztergombi|first5=K.|date=2008-12-20|title=Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing|url=https://www.sciencedirect.com/science/article/pii/S0001870808002053|journal=Advances in Mathematics|language=en|volume=219|issue=6|pages=1801–1851|doi=10.1016/j.aim.2008.07.008|issn=0001-8708}}</ref> <math>W</math> to <math>W_G</math>, and see <math>t(F, W)</math> as roughly the fraction of labeled copies of graph <math>F</math> in "approximate" graph <math>G</math>. Then, we can assume the quantity <math>t(F, W) + t(F, 1 - W)</math> is roughly <math>t(F, G) + t(F, \overline{G})</math> and interpret the latter as the combined number of copies of <math>F</math> in <math>G</math> and <math>\overline{G}</math>. Hence, we see that <math>t(F, G) + t(F, \overline{G}) \gtrsim 2^{-e(F)+1}</math> holds. This, in turn, means that common graph <math>F</math> commonly appears as subgraph.

In other words, if we think of edges and non-edges as [[Edge coloring|2-coloring of edges]] of complete graph on the same vertices, then at least <math>2^{-e(F)+1}</math> fraction of all possible copies of <math>F</math> are monochromatic. TheNote abovethat definitionin usinga the[[Erdős–Rényi generalizedmodel|Erdős–Rényi random graph]] <math>G = G(n, p)</math> with each edge drawn with probability <math>p=1/2 </math>, each [[graph homomorphism]] densityfrom can<math>F</math> beto understood<math>G</math> inhave thisprobability way<math>2 \cdot 2^{-e(F)} = 2^ {-e(F) +1}</math>of being monochromatic. So, common graph <math>F</math> is a graph where it attains its minimum number of appearance as a monochromatic subgraph of graph <math>G</math> at the graph <math>G=G(n, p)</math> with <math>p=1/2</math>

<math>p=1/2</math>. 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 [[Sidorenko's conjecture#Partial results|known Sidorenko graph]] is an example of a common graph, and, most notably, [[Cycle (graph theory)|cycles of even length]] are common<ref>{{Cite journal|last=Sidorenko|first=A. F.|date=1992|title=Inequalities for functionals generated by bipartite graphs|url=https://www.degruyter.com/document/doi/10.1515/dma.1992.2.5.489/html|journal=Discrete Mathematics and Applications|volume=2|issue=5|doi=10.1515/dma.1992.2.5.489|issn=0924-9265}}</ref>.However, these are limited examples since all Sidorenko graphs are [[Bipartite graph|bipartite graphs]] while there exist non-bipartite common graphs, as demonstrated below.

* The [[triangle graph]] <math>K_{3}</math> is aone simple example of non-bipartite common graph.

* <math>K_4 ^{-}</math>, the graph obtained by removing an edge of the [[complete graph]] on 4 vertices <math>K_4</math>, is common.

* Non-example: It was believed for a time that all graphs are common. However, Thomasonshockingly, provedit turns out that <math>K_{t}</math> is not common for <math>t \ge 4</math>, as proved by Thomason in 1989.<ref>{{Cite journal|last=Thomason|first=Andrew|date=1989|title=A Disproof of a Conjecture of Erdős in Ramsey Theory|url=https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-39.2.246|journal=Journal of the London Mathematical Society|language=en|volume=s2-39|issue=2|pages=246–255|doi=10.1112/jlms/s2-39.2.246|issn=1469-7750}}</ref> In particular, <math>K_4</math> is not common even though <math>K_{4} ^{-}</math> is common.

== Proofs ==

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

===Sidorenko graphs are common===

Recall that a Sidorenko graph <math>F</math> is a graph satisfying <math>t(F, W) \ge t(K_2, W)^{e(F)}</math> for all graphons <math>W</math>. Hence, we should also have <math>t(F, 1 - W) \ge t(K_2, 1 - W)^{e(F)}</math>. Now, observe that <math>t(K_2, W) + t(K_2, 1 - W) = 1 </math>, which follows from the definition of homomorphism density. Combining this with [[Jensen's inequality]] for the function <math>f(x) = x^{e(F)}</math>, we can see that

Recall that a Sidorenko graph <math>F</math> is a graph satisfying <math>t(F, W) \ge t(K_2, W)^{e(F)}</math> for all graphons <math>W</math>. Hence, we should also have <math>t(F, 1 - W) \ge t(K_2, 1 - W)^{e(F)}</math>. Now, observe that <math>t(K_2, W) + t(K_2, 1 - W) = 1 </math>, which follows from the definition of homomorphism density. Combining this with [[Jensen's inequality]] for the function <math>f(x) = x^{e(F)}</math>, we can see that

<math>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}</math>

Thus, the conditions for common graph is met.

===The triangle graph is common===

Here, we will expand the integral expression for <math>t(K_3, 1 - W)</math> and take into account the symmetry between the variables:

<math>\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</math>

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:

:<math>\int_{[0, 1]^2} W(x, y) dx dy = t(K_2, W) </math>

: <math>\intint_{[0, 1]^32} W(x, y) W(x, z) dx dy dz = t(K_{1, 2}K_2, W) </math>

: <math>\int_int{[0, 1]^3} W(x, y) W(yx, z) W(z, x) dx dy dz = t(K_3K_{1, 2}, W) </math>

: <math>\int_{[0, 1]^23} W(x, y) W(y, z) W(z, x) dx dy dz = t(K_2K_3, W) </math>

(Note that <math>K_{1, 2}</math> denotes the [[complete bipartite graph]] on <math>1</math> vertex on one part and <math>2</math> vertices on the other.) Hence, we get:

:<math>t(K_3, W) + t(K_3, 1 - W) = 1 - 3 t(K_2, W) + 3 t(K_{1, 2}, W) </math>.

Now, in order to relate: <math>t(K_{1, 2}K_3, W)</math> to+ <math>t(K_2K_3, 1 - W)</math>, note= that1 we- can3 exploitt(K_2, theW) symmetry+ between3 thet(K_{1, variables2}, <math>yW) </math> and <math>z</math> to write:.

Now, in order to relate <math>t(K_{1, 2}, W)</math> to <math>t(K_2, W)</math>, note that we can exploit the symmetry between the variables <math>y </math> and <math>z</math> to write:<math display="block">\begin{alignat}{4}

<math display=block>\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}</math>where we used the integral [[Cauchy–Schwarz inequality]] in the last step. Finally, our desired result follows from the above inequality:

\end{alignat}</math>

where we used the integral [[Cauchy–Schwarz inequality]] in the last step. Finally, our desired result follows from the above inequality:

<math>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</math>

== See also ==