Common graph: Difference between revisions

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{{Short description|InConcept in extremal graph theory, common graphs arise as an interesting concept}}

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{{AFC comment|1=Good coverage in ''Large Networks and Graph Limits'', but I'd still like to see references inline for each piece of information. — [[User:Bilorv|Bilorv]] ('''[[User talk:Bilorv|<span style="color:purple">talk</span>]]''') 20:07, 20 January 2022 (UTC)}}

{{AFC comment|1=Really interesting read, but the Wikipedia style isn't there yet. I've gone through rewriting and formatting to be more in line with the Wikipedia style, but the referencing remains to be done.

We need more papers than this that mention/use/introduce theory around common graphs, and every fact in Wikipedia needs to be clearly attributed to a particular source. Which reference does each proof come from? Where does the definition, the intuitions and the examples come from? — [[User:Bilorv|Bilorv]] ('''[[User talk:Bilorv|<span style="color:purple">talk</span>]]''') 01:48, 22 December 2021 (UTC)}}

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{{Short description|In extremal graph theory, common graphs arise as an interesting concept}}

{{Draft topics|computing}}

{{AfC topic|stem}}

In [[graph theory]], an area of mathematics, '''common graphs''' belong to a branch of [[extremal graph theory]] concerning [[Homomorphism density|inequalities in homomorphism densities]]. Roughly speaking, <math>F</math> is a common [[Graph (discrete mathematics)|graph]] if it "commonly" appears as a subgraph, in a sense that the total number of copies of <math>F</math> in any graph <math>G</math> and its [[Complement graph|complement]] <math>\overline{G}</math> is a large fraction of all possible copies of <math>F</math> on the same vertices. Intuitively, if <math>G</math> contains few copies of <math>F</math>, then its complement <math>\overline{G}</math> must contain lots of copies of <math>F</math> 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 conjecture#Statement|Sidorenko's propertygraphs]]).

== Definition ==

Formally, a common graph is aA graph <math>F</math> suchis common thatif 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 webbook|title=Large Networks and Graph Limits|url=https://bookstore.ams.org/coll-60/|access-date=2022-01-13|websitepublisher=bookstore.ams.org}}</ref>American and a survey "''Very Large Graphs''"<ref>{{CiteMathematical journalSociety|lastpage=Lovasz|first=Laszlo|date=2009-02-01|title=Very large graphs|url=http://arxiv.org/abs/0902.0132|journal=arXiv:0902.0132 [math]297}}</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.       

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

== 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|lastlast1=Borgs|firstfirst1=C.|last2=Chayes|first2=J. T.|last3=Lovász|first3=L.|authorlink3=László Lovász|last4=Sós|first4=V. T.|authorlink4=Vera T. Sós|last5=Vesztergombi|first5=K.|authorlink5=Katalin Vesztergombi|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|doi-access=free|s2cid=5974912|issn=0001-8708|arxiv=math/0702004}}</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. Note that in a [[Erdős–Rényi model|Erdős–Rényi random graph]] <math>G = G(n, p)</math> with each edge drawn with probability <math>p=1/2 </math>, each [[graph homomorphism]] from <math>F</math> to <math>G</math> have probability <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>

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== Examples ==

* As stated above, all Sidorenko graphs are common graphs.<ref>{{Cite book|title=Large Networks and Graph Limits|url=https://bookstore.ams.org/coll-60/|access-date=2022-01-13|publisher=American Mathematical Society|page=297}}</ref> 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|s2cid=117471984|issn=0924-9265|url-access=subscription}}</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 one simple example of non-bipartite common graph.<ref>{{Cite book|title=Large Networks and Graph Limits|url=https://bookstore.ams.org/coll-60/|access-date=2022-01-13|publisher=American Mathematical Society|page=299}}</ref>

* <math>K_4 ^{-}</math>, the graph obtained by removing an edge of the [[complete graph]] on 4 vertices <math>K_4</math>, is common.<ref>{{Cite book|title=Large Networks and Graph Limits|url=https://bookstore.ams.org/coll-60/|access-date=2022-01-13|publisher=American Mathematical Society|page=298}}</ref>

* Non-example: It was believed for a time that all graphs are common. However, shockingly, it 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|url-access=subscription}}</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 SidorenkoA graph <math>F</math> is a Sidorenko graph satisfyingif it satisfies <math>t(F, W) \ge t(K_2, W)^{e(F)}</math> for all graphons <math>W</math>.

In Hencethat case, we should also have <math>t(F, 1 - W) \ge t(K_2, 1 - W)^{e(F)}</math>. NowFurthermore, 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. This proof, along with proof of <math>K_4^{-}</math> being common graph, can be seen from page 297-298 of the aforementioned book of László Lovász<ref>{{Cite book|last=Lovász|first=László|title=Large Networks and Graph Limits|publisher=American Mathematical Society Colloquium publications|year=2012|isbn=978-0821890851|___location=United States|pages=297-298297–298|language=English}}</ref>.

===The triangle graph is common===

Here, we will expandExpand 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 eachEach term in the expression can be written in terms of homomorphism densities of smaller graphs. Indeed, byBy the definition of homomorphism densities, we have:

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

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: <math>\int_{[0, 1]^3} W(x, y) W(y, z) W(z, x) dx dy dz = t(K_3, W)</math>

(Note thatwhere <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, weIt getfollows:

: <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}, W)</math> can be related to <math>t(K_2, W)</math>, notethanks that we can exploitto the symmetry between the variables <math>y </math> and <math>z</math> to write:
<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>

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

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

<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>.

This above proof can be obtained from taking the continuous analog of Theorem 1 in Goodman 1959 paper, "On Sets Of Acquaintances And Strangers At Any Party"<ref>{{Cite journal|last=Goodman|first=A. W.|date=1959|title=On Sets of Acquaintances and Strangers at any Party|url=https://www.jstor.org/stable/2310464|journal=The American Mathematical Monthly|volume=66|issue=9|pages=778–783|doi=10.2307/2310464|jstor=2310464|issn=0002-9890|url-access=subscription}}</ref>.

== See also ==

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== References ==

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{{reflist}}

[[Category:Graph families]]

[[Category:Extremal graph theory]]