On the diameter of random planar graphs

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1 On the dameter of random planar graphs Gullaume Chapuy, CNRS & LIAFA, Pars jont work wth Érc Fusy, Pars, Omer Gménez, ex-barcelona, Marc Noy, Barcelona. Probablty and Graphs, Eurandom, Endhoven, 04.

2 Planar graphs and maps Planar graph = (connected) graph on V = {,,..., n} that can be drawn n the plane wthout edge crossng.

3 Planar graphs and maps Planar graph = (connected) graph on V = {,,..., n} that can be drawn n the plane wthout edge crossng. Planar map = planar graph + planar drawng of ths graph (up to contnuous deformaton) = same graph dfferent maps

4 Planar graphs and maps Planar graph = (connected) graph on V = {,,..., n} that can be drawn n the plane wthout edge crossng. Planar map = planar graph + planar drawng of ths graph (up to contnuous deformaton) = same graph dfferent maps Note: the number of embeddngs depends on the graph... Unform random planar map Unform random planar graph!

5 Some known results for maps (stated approxmately) Thm [Chassang-Schaeffer 04], [Marckert, Mermont 06], [Ambjörn-Budd ] In a unform random map M n of sze n, dstances are of order n /4. For example one has Dam(M n) some real random varable n /4 n /4 c J.-F. Marckert

6 Some known results for maps (stated approxmately) Thm [Chassang-Schaeffer 04], [Marckert, Mermont 06], [Ambjörn-Budd ] In a unform random map M n of sze n, dstances are of order n /4. For example one has Dam(M n) some real random varable n /4 n /4 c J.-F. Marckert

Some known results for maps (stated approxmately) Thm [Chassang-Schaeffer 04], [Marckert, Mermont 06], [Ambjörn-Budd ] In a unform random map M n of sze n, dstances are of order n /4.

7 Some known results for maps (stated approxmately) Thm [Chassang-Schaeffer 04], [Marckert, Mermont 06], [Ambjörn-Budd ] In a unform random map M n of sze n, dstances are of order n /4. For example one has Dam(M n) some real random varable n /4 n /4 c J.-F. Marckert A lot of (very strong) thngs are known very actve feld of research snce 004 [Boutter, D Francesco, Gutter, Le Gall, Mermont, Pauln, Addaro-Berry, Albenque...]

8 Our man result: dameter of random planar GRAPHS Thm [C, Fusy, Gménez, Noy 00+] Let G n be the unform random planar graph wth n vertces. Then Dam(G n ) = n /4+o() w.h.p. ( More precsely P Dam(G n ) [ n /4 ɛ, n /4+ɛ] ) = O(e nθ(ɛ) ).

9 Our man result: dameter of random planar GRAPHS Thm [C, Fusy, Gménez, Noy 00+] Let G n be the unform random planar graph wth n vertces. Then Dam(G n ) = n /4+o() w.h.p. ( More precsely P Dam(G n ) [ n /4 ɛ, n /4+ɛ] ) = O(e nθ(ɛ) ). Ths s some knd of large devaton result. We also conjecture convergence n law: Dam(G n ) some real random varable n /4 Note: for random trees, Dam(T n ) n / some real random varable ( P Dam(T n ) [ n / ɛ, n /+ɛ] ) = O(e nθ(ɛ) ) [Flajolet et al 9]

10 (0) Connectvty n graphs General A graph s k-connected f one needs to remove at least k vertces to dsconnect t. Connected (-connected) -Connected -Connected

11 (0) Connectvty n graphs General A graph s k-connected f one needs to remove at least k vertces to dsconnect t. Connected (-connected) maps graphs -Connected -conn. maps -conn. graphs -Connected -conn. maps same thng [Tutte 66]: - a connected graph decomposes nto -connected components - a -connected graph decomposes nto -connected components [Whtney]: A -connected planar graph has a UNIQUE embeddng -conn. graphs

12 (0) Connectvty n graphs General A graph s k-connected f one needs to remove at least k vertces to dsconnect t. Connected (-connected) maps graphs -Connected -conn. maps -conn. graphs -Connected -conn. maps same thng [Tutte 66]: - a connected graph decomposes nto -connected components - a -connected graph decomposes nto -connected components [Whtney]: A -connected planar graph has a UNIQUE embeddng -conn. graphs [Tutte 60s], [Bender,Gao,Wormald 0], [Gménez, Noy 05] followed ths path carryng countng results along the scheme exact countng of planar graphs! Here we follow the same path and carry devatons statements for the dameter.

13 maps () graphs -conn. maps -conn. graphs -conn. maps -conn. graphs

14 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex 0

15 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte)

16 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

17 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

18 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

19 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

20 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

21 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules:

22 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules: Fact: the blue map s a tree. +

23 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules: Fact: the blue map s a tree. +

24 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules: Fact: the blue map s a tree. +

25 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules: Fact: the blue map s a tree. +

26 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) To smplfy the exposton we consder a quadrangular planar map (faces have degree 4). Label vertces by ther graph-dstance to some root vertex. Observe that there are only two types of faces (snce bpartte). Apply Schaeffer rules: Fact: the blue map s a tree. + If one remembers the labels the constructon s bjectve!

27 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) A well-labelled tree s a plane tree together wth a mappng l : V Z >0 such that - f v v then l(v) l(v ) - mn v l(v) = 4 4 4

28 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) A well-labelled tree s a plane tree together wth a mappng l : V Z >0 such that - f v v then l(v) l(v ) - mn v l(v) = Thm [Cor-Vauqueln 8;Schaeffer 99] There s a bjecton between quadrangular planar maps wth a ponted vertex and n + vertces and well-labelled trees wth n vertces. The labels n the tree correspond to dstances to the root n the map

29 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) A well-labelled tree s a plane tree together wth a mappng l : V Z >0 such that - f v v then l(v) l(v ) - mn v l(v) = Thm [Cor-Vauqueln 8;Schaeffer 99] There s a bjecton between quadrangular planar maps wth a ponted vertex and n + vertces and well-labelled trees wth n vertces. The labels n the tree correspond to dstances to the root n the map. Corollary: Dam(M n ) = n /4+o() ndeed: - the heght of a random tree s = n /+o() w.h.p n /

30 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) A well-labelled tree s a plane tree together wth a mappng l : V Z >0 such that - f v v then l(v) l(v ) - mn v l(v) = Thm [Cor-Vauqueln 8;Schaeffer 99] There s a bjecton between quadrangular planar maps wth a ponted vertex and n + vertces and well-labelled trees wth n vertces. The labels n the tree correspond to dstances to the root n the map. Corollary: Dam(M n ) = n /4+o() ndeed: - the heght of a random tree s = n /+o() w.h.p - the labellng functon behaves as a random walk along branches of the tree so l(v) n /+o() = n /4+o() n /

31 () Maps: the Cor-Vauqueln-Schaeffer bjecton ( ) A well-labelled tree s a plane tree together wth a mappng l : V Z >0 such that - f v v then l(v) l(v ) - mn v l(v) = Thm [Cor-Vauqueln 8;Schaeffer 99] There s a bjecton between quadrangular planar maps wth a ponted vertex and n + vertces and well-labelled trees wth n vertces. The labels n the tree correspond to dstances to the root n the map. Corollary: Dam(M n ) = n /4+o() ndeed: - the heght of a random tree s = n /+o() w.h.p - the labellng functon behaves as a random walk along branches of the tree so l(v) n /+o() = n /4+o() [Chassang-Schaeffer 04] n /

32 maps -conn. maps () graphs -conn. graphs -conn. maps -conn. graphs

33 () Decomposton nto -connected components A connected map s a tree of -connected maps

34 () Decomposton nto -connected components A connected map s a tree of -connected maps One can wrte ths n terms of generatng functons.

35 () Decomposton nto -connected components A connected map s a tree of -connected maps One can wrte ths n terms of generatng functons. The generatng functon of connected maps s explctly known (e.g. usng bjectons) deduce the g.f. of -connected maps from the one of connected maps. [Tutte 60 s].

36 () Decomposton nto -connected components A connected map s a tree of -connected maps One can wrte ths n terms of generatng functons. The generatng functon of connected maps s explctly known (e.g. usng bjectons) deduce the g.f. of -connected maps from the one of connected maps. [Tutte 60 s].

37 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] n + O(n/ ) O(n / )

38 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] tools to prove ths: (very) fne sngularty analyss of generatng functons. In partcular exact expressons for countng functons are mandatory! n + O(n/ ) O(n / ) H H H X n

39 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] tools to prove ths: (very) fne sngularty analyss of generatng functons. In partcular exact expressons for countng functons are mandatory! n + O(n/ ) O(n / ) Corollary : Dam(B n/ ) Dam(M n ) = n /4+o() random -conn. map of sze n/ H H H X n

40 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] tools to prove ths: (very) fne sngularty analyss of generatng functons. In partcular exact expressons for countng functons are mandatory! n + O(n/ ) O(n / ) Corollary : Dam(B n/ ) Dam(M n ) = n /4+o() random -conn. map of sze n/ H H H X n ndeed: Dam(X n ) Dam(M n ) Dam(X n ) + max Dam(H )

41 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] tools to prove ths: (very) fne sngularty analyss of generatng functons. In partcular exact expressons for countng functons are mandatory! n + O(n/ ) O(n / ) Corollary : Dam(B n/ ) Dam(M n ) = n /4+o() random -conn. map of sze n/ ndeed: Dam(X n ) Dam(M n ) Dam(X n ) + max Dam(H ) H H H X n ( n /) /4+ɛ w.h.p.

42 () Decomposton nto -connected components Thm The largest -connected component has sze n + n/ A where A converges to an explct law. The second-largest component has sze O(n / ). [Gao, Wormald 99] [Banderer, Flajolet, Schaeffer, Sora 0] tools to prove ths: (very) fne sngularty analyss of generatng functons. In partcular exact expressons for countng functons are mandatory! n + O(n/ ) O(n / ) Corollary : Dam(B n/ ) Dam(M n ) = n /4+o() random -conn. map of sze n/ ndeed: Dam(X n ) Dam(M n ) Dam(X n ) + max Dam(H ) and X n s essentally a random -conn. map of sze n/. H H H X n ( n /) /4+ɛ w.h.p.

43 maps graphs -conn. maps -conn. graphs -conn. maps -conn. graphs

44 maps graphs -conn. maps -conn. maps () -conn. graphs -conn. graphs

45 () Decomposton nto -connected components T R M -connected T T R R Agan one can wrte everythng n terms of generatng functons. deduce the g.f. of -conn. maps from the one of -connected maps. [Tutte 60 s]. deduce the g.f. of -conn. graphs from the one of -connected graphs [Bender, Gao, Wormald 0]. T = -connected component R = seres composton M = parallel composton

46 () Decomposton nto -connected components Prop A random -connected planar graph wth n edges has dameter n /4+o() wth hgh probablty. R M T RMT tree a -conn. graph B n T T R R

47 () Decomposton nto -connected components Prop A random -connected planar graph wth n edges has dameter n /4+o() wth hgh probablty. R M T RMT tree T a -conn. graph B n Same dea: - there exsts a T -component Y n of lnear sze w.h.p. T R R

48 () Decomposton nto -connected components Prop A random -connected planar graph wth n edges has dameter n /4+o() wth hgh probablty. R M T RMT tree T T R R a -conn. graph B n Same dea: - there exsts a T -component Y n of lnear sze w.h.p. - the dameter of the RMT-tree s n o() w.h.p. - The extra-length due the edge substtuton s also n o()

49 Concluson (I) Thm [C, Fusy, Gménez, Noy 00+] Let G n be the unform random planar graph wth n vertces. Then Dam(G n ) = n /4+o() w.h.p. ( More precsely P Dam(G n ) [ n /4 ɛ, n /4+ɛ] ) = O(e nθ(ɛ) ). The proof reles both on exact generatng functons and magcal bjectons: we couldn t do anythng wthout ths (or maybe somethng much weaker lke O( n)?) The general pcture s qute clear but the analyss s a bt tedous... (need to work wth bvarate generatng functons and prove estmates wth enough unformty) No way to obtan the convergence of Dam(G n ) - even for planar maps ths s very n /4 dffcult! Same result for the unform random graph wth n vertces and µn edges for < µ <.

50 Concluson (II) We generalzed the Gménez-Noy enumeraton result to graphs embeddable on a surface of genus g 0 Thm [C, Fusy, Gménez, Mohar, Noy 0] [Bender-Gao 0] #{n-vertex genus g graphs} c g n! γ n n 5 g 7/ γ Same knd of proof but Whtney s maps theorem (unqueness of embeddng) now requres that there s no short non-contractble cycle. -conn. (but we could prove that) maps The result on the dameter should be the same but ths s not (and won t be) wrtten. -conn. maps The fact that non-contractble cycles are small mply the followng: Thm [C, Fusy, Gménez, Mohar, Noy 0] Fx g. The random graph of genus g and sze n has chromatc number n {4, 5} and lst chromatc number 5 w.h.p. graphs -conn. graphs -conn. graphs

51 Thank you!

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