Spanning Maximal Planar Subgraphs of Random Graphs

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1 Spaig Maximal Plaar Subgraphs of Radom Graphs 6. Bollobiis* Departmet of Mathematics, Louisiaa State Uiversity, Bato Rouge, LA A. M. Frieze? Departmet of Mathematics, Caregie-Mello Uiversity, Pittsburgh, PA ABSTRACT We study the threshold for the existece of a spaig maximal plaar subgraph i the radom graph G,p. We show that it is very ear p = &. We also discuss the threshold for the existece of a spaig maximal outerplaar subgraph. This is very ear p = A. Key Words: radom graphs, plaar graphs. 1. INTRODUCTION I this short ote we codider the threshold for the property that the radom graph G,p cotais a spaig maximal plaar subgraph, i.e., a plaar subgraph with 3-6 edges ad 2-4 triagular faces. Our otatio ad termiology follows [l]; i particular G,p is the radom graph with vertex set [] = { 1,2,..., } which is obtaied by selectig each of the N = ( ) possible edges idepedetly, with probability p. Let us defie the graph property d by settig G = (V, E) E d if E cotais a set F of 31VI - 6 edges such that (V, F) is plaar. Thus G E d iff G cotais a maximal plaar graph spaig the etire graph. * Research supported i part by NSF Grat DMS t Research supported i part by NSF Grat CCR Radom Structures ad Algorithms, Vol. 2, No. 2 (1991) Joh Wiley & Sos, Ic. CCC /91/ $ s

2 226 BOLLOBAS AND FRIEZE Theorem 1. C (a) Let p = 3 where c = (27e/256)'I3. The a.e. G,,,#d. (b) Let p = c(log )l13 where CP 100. The a.e. G,,, E d. 113 Note the small gap withi which the exact threshold has bee located. It is difficult to speculate what the exact threshold value is. Note also that the simplest "local" coditio that every vertex lies o at least oe triagle is ot almost always sufficiet, i cotrast with may other graphs properties (see [l]). Note also that -k'(3k-6) is the exact threshold for cotaiig ay fixed maximal plaar subgraph with k vertices. The techiques used to prove Theorem 1 ca be modified to prove aother problem. Recall that a graph is outerplaar if it ca be draw o the plae with every vertex icidet with the outer face. A maximal outerplaar graph is oe i which every face other the the outer oe is a triagle. A -vertex maximal outerplaar graph has 2-3 edges. Let B deote the property of cotaiig a maximal outerplaar graph spaig the etire graph. Theorem 2. C (a) Let p = 1/3 where c = (e/4)'12. The a.e. G,,, #B. (b) Let p = '(log )1'2 where c > S fi. The a.e. G,,, E 3. 'l' The proofs of these two theorems are give i the ext two sectios. 2. PROOF OF THEOREM 1 Let M, be the umber of maximal plaar subgraphs with labelled vertices. The P(G,,, E d) 5 M,P~"-~. As almost every maximal plaar graph has a trivial automorphism group, Tutte's classical formula [4] implies that if is sufficietly large (i fact, for all ). Hece, if p = 113 where c = (27e/256)l13, the P(G,,, E d) < 1 / ad this proves part (a). C

3 SPANNING PLANAR SUBGRAPHS 227 Fig. 1 1 To prove part (b) we eed to defie some specific triagulatios that we ca costruct with high probability. Let T = T, be the 1Pvertex triagulatio of Figure 1. Note that T, ca be costructed from the outer triagle by a sequece of vertex isertios. By this we mea take a face F = xyz ad the isert a ew vertex u ito F by addig edges ux, uy, uz. Thus we ca start with outer triagle, isert the vertex labelled 2, isert the vertices labelled 3 ito 3 of the faces ad so o. We refer to these isertios as operatios 2, 3, 4, ad 5. T, is the first i a sequece T,, T,,..., T,,.... Costruct T, from T, by isertig a copy of T, ito each of the 6 special faces labelled 3, 4, 5. After isertio the vertices iside each special face are umbered as they are i TI ad so T, has 36 special faces. I geeral T, is obtaied from Tk-] by isertig a copy of T, ito each special face ad umberig the vertices as above. T, has the followig statistics: (i) 6, special faces; (ii) tk = 4( 16-6, - 1) vertices; (iii) maximum degree 18. (We obtai (ii) from the recurrece t, = fk-1 + gk ) Now defie Tk,i, i = 0,1,2,3,4 as follows: Tk,o = Tk ad Tk,i is obtaied from Tk,i-l by applyig Operatio i + 1 to those subgraphs cotaied i each of what was a special face of Tk. Thus T,,., = Tk+,. It is coveiet to let To deote a triagle. c(log )1 3 Suppose ow that p = 113 where c = 100. Let p1 satisfy 1 - (1- p1)l0 = p so that p, >p/lo. We ca assume that G,p is the uio of 10 idepedet copies of G,+,. Let E,, El, E,,..., E9 deote the edge sets of these copies.

4 228 BOLLOBAS AND FRIEZE Let k, = max{ k : 2tk } = [log,( ( )) J We try to costruct a spaig maximal plaar subgraph of G.p as follows: we will fill i the details of each step of the costructio later. A: costruct To usig edges i E,; B: fork=otok,-1 do for j = 1 to 4 do costruct a copy of Tk,j from the copy of Tk,j-l via operatio j ad usig oly edges from E,, r = ((k + l)j mod 8) + 1 C: augmet the copy of Tk, to a spaig triagulatio by vertex isertio usig edges from E, oly. We must ow show that we ca complete the costructio above with probability 1 - o( 1). A: G, has a triagle with high probability if w = w()+ w. Sice p, + m we,; ca be sure that A succeeds with probability 1 - o(1). B: the process of costructig Tk,j from Tk,j-l ivolves tryig to isert a vertex ito each of at most $ triagles. Suppose that the vertices outside of our copy of Tk,j-l comprise Vl(k, j) ad the vertices of the triagles ito which we are tryig to isert vertices from Vl(k, j) comprise V,(k, j) V,(k - 1, j). We are examiig edges from E,. The previous time we used edges from E,, the vertices i V,(k, j) were outside of the the curret triagulatio Tk-2,j- ad so the E, edges betwee V,(k, j) ad V,(k, j) are ucoditioed by the history of the costructio to this poit. To show that Tk,j ca almost always be costructed from Tk,j-l we defie a bipartite graph BP(k, j) with vertex partitio V,(k, j) ad S(k, j) = {faces F of Tk,j-, ito which a ew vertex is iserted i the creatio of Tk,j}. BP(k, j) has a edge uf wheever u E Vl(k, j) is adjacet i G, = ([], E,) to all vertices of FE S(k, j). Note that P(uFE E,) =p: but that these edges do ot appear idepedetly. To complete the aalysis of B, we eed oly prove that (1) P(BP(k, j) cotais a matchig of size )S(k, j)l) = 1.- o((1og )-'). Because the edges of BP(k, j) do ot appear,idepedetly we agai resort to the trick of partitioig the edge set. Let E, =,U Er,i where the edges of Er,i are chose idepedetly with probability p,, 1 - (1 - p,)' = pl, p2 2 y. Cosider the graph T(k, j) which has vertex set S(k, j) ad a edge F,F,, where F,, F, E S(k, j), wheever Fl ad F, share a vertex i T(k, j - 1). It is ot hard to see that the maximum vertex degree i T(k, j) is at most 7 (whe j = 3 ad accoutig for special faces sharig a vertex.) It is therefore possible to color these triagles usig oly 7 colors so that tri5gles of the same color are vertex 1=l

5 SPANNING PLANAR SUBGRAPHS 229 disjoit. Let us ow decompose BP(k, j) as.u BP(k, j, i) where BP(k, j, i) has the i-colored triagles deoted S(k, j, i) C S(i,'j), all of Vl(k, j) ad a edge uf if u is adjacet to all vertices of F via edges of color Er,i. Edges i BP(k, j, i) ow appear idepedetly with probability pi > p3/703. We ca ow use the result of Erdos ad RCyi [2] (see also [l, pp ) cocerig the threshold for a perfect matchig i a radom bipartite graph. Actually, we oly eed a matchig from S(k, j, i) to V,(k, j). By choice of k,, IVl(k, j + 1)1? - always ad - p3. > log, so we ca first match S(k, j, i) to a subset V,(k, j, 1) of Vl(k, j) ad the S(k, j, 2) to a subset of V,(k, j)\v,(k, j, 1) ad so o, with sufficietly high probability (observe that the domiat failure probability i Erdos ad RCyi's result comes from isolated vertices). This completes the aalysis of B. C: sice the maximum degree i Tk, is 18, each face of Tk, shares a vertex with at most 51 other faces. Also Tk, has at least - faces ad so it is 12 possible to fid - vertex disjoit faces i Tk,. Furthermore, if a triagula- 612 tio cotais (Y vertex disjoit faces ad a vertex is iserted ito oe of these faces, the the ew triagulatio has at least (Y vertex disjoit faces. Let u,, u2,..., u,, m < be a eumeratio of the vertices outside of Tk,. We will try to isert ui, i = 1,2,..., m sequetially ito the curret triagulatio usig edges i E9 oly. Sice there are always at least - vertex disjoit faces 612 available, we have 3 /612 P( 3i : ui caot be iserted) 5 ( 1 - p,) < -112 ad this shows that we ca complete the costructio with high probability ad completes the proof of Theorem PROOF OF THEOREM 2 The proof of part (a) is similar that of part (a) of Theorem 1. Let O be the umber of maximal outerplaar graphs with labelled vertices. The P(G,, E 93) 5 0,,p2-3. But it is kow (e.g., Lovhz [3, Problem 39 of Ch. 11) that ( 2)! (2-4 ) <- ( 4;)-2 o ad so if p = (e/4)'/', the ad this proves part (a). C2-3 P(G,,, E 93) 5-3/2 O < -'l2

6 230 BOLLOBAS AND FRIEZE For part (b) we provide a costructio which ca be show to work with high probability. We oce agai assume that G,p is give as the uio of a umber of idepedet copies of G,pl. Here four will suffice, so that 1 -(1- p1)4 = p ad p1 P 2 -. Let E,, El, E,, E, deote the edge sets of these copies. 4 A: costruct a triagle A, usig edges of E, oly. B: fork=otok,=[log,~]-l do At this poit A, is a outerplaar subgraph of G,p cotaiig, = 3-2, vertices. Let the edges of the outer face of A, be el, e2,..., ek where ei, ei+l are adjacet i = 1,2,...,, - 1. Let Fl = {el, e,, e,,...} ad F, = {e2, e4, e6,...} be the odd ad eve idexed edges, respectively. for j = 1 to 2 do costruct the bipartite graph BP (k, j) with vertices 5 U ([]\V(A,,j-l)) [A,,, = A,, A,,l is costructed durig j = 1 ad A,,2 = There is a edge ue, U~ZV(A,,~-~), e q wheever u is adjacet to both edpoits of e by edges i Ej. If BP (k, j) cotais a matchig of size 151, the we ca use this matchig to add 151 vertices to Ak,j-l i a obvious way. See Figure 2. C: augmet Akl to a spaig maximal idepedet outerplaar graph usig edges from E, oly. We must ow show that we ca complete the costructio above with probability 1 - o(1). A: as for A i the previous sectio. B: the edges of BP (k, j) occur idepedetly with probability p2 =- PZ -. We -16 ca apply the result of Erdos ad RCyi as before sice lv(a,,j-l)l 5 - by 2 defiitio of k 1. C: let ul, u2,..., u, be a eumeratio of []\V(Akl). For i = 1,2,..., m we try to fid a edge o the outerface of the curret triagulatio for which both edpoits are adjacet to ui usig edges i E,. Sice V(A,,) has at Fig. 2

7 SPANNING PLANAR SUBGRAPHS 231 least - vertices ad the E3 edges icidet with ui are ucoditioed by the 4 previous history we have P(3i : ui caot be added) 5 ( 1 - p:) 4 = o(1) for c >8fi This completes the proof of Theorem FINAL REMARKS The reader will observe that the costats i parts (b) of the theorems ca easily be reduced, but that it is ot clear how to icrease those i parts (a). The mai questio left ope by this paper is the whereabouts of the exact thresholds. Oe ca also ask for the threshold for the existece of spaig plaar subgraphs with a ed es, a>l. The argumet of parts (a) shows that the threshold is at least - ad it seems likely that the costructive method we have used ca be adapted to attack this problem. REFERENCES [l] B. Bollobh, Radom Graphs, Academic Press, Orlado, FL, 1985, xvi + 447pp. [2] P. Erdos ad A. RCyi, O radom matrices, Publ. Math. Zst. Hugar. Acad. Sci., 8, (1964). [3] L. Lovbz, Combitorial Problems ad Exercises, North-Hollad, Amsterdam, New York, Oxford, 1979, 551 pp. [4] W. T. Tutte, A cesus of plaar triagulatios, Caad. J. Math., 14, (1962). Received February 16, 1990