ON SIZE RAMSEY NUMBERS OF GRAPHS WITH BOUNDED DEGREE VOJT ECH R ODL AND ENDRE SZEMER EDI Abstract. Aswerig a questio of J. Beck [B2], we prove that there exists a graph G o vertices with maximum degree three ad the size Ramsey umber ^r(g) c(log ) where ad c are positive costats. For graphs G ad F, write F! G to mea that if the edges of F are colored by red ad blue, the F cotais a moochromatic copy of G. Erd}os, Faudree, Rousseau ad Schelp [EFRS] were the rst to cosider the questio of how few edges F ca have ad still F! G. Followig [EFRS], by the size Ramsey umber ^r(g), we mea the least iteger ^r such that there exists a graph F with ^r edges for which F! G, i. e., ^r(g) = mifjf j : F! Gg (where jf j deotes the cardiality of the edge set of F ). Clearly, ^r(k 1; ) = 2?1, where K 1; deotes the star o +1 vertices. I [B1], J. Beck aswered a questio of P. Erd}os [E], provig that there exists a absolute costat c (c 900) such that (1) ^r(p) < c: Later i [B2], J. Beck raised the followig questio, which if aswered armatively, would give a far reachig geeralizatio to (1). Problem. Let G;r be a graph with vertices ad maximum degree r. Decide whether ^r(g;r) < c(r) where the costat c(r) depeds oly o r. This has show to be true whe G;r is a cycle [HKL] or a tree [FP]. The aim of this ote is to aswer Beck's questio showig that the statemet above fails already for r = 3. More precisely, we prove the followig theorem: Theorem 1. There exists positive costats c ad, ad a graph G = (V; E) with jv j = ad maximum degree, (G) = 3 such that (2) ^r(g) c( ) : Proof: Sice we believe that the lower boud (2) is quite far from the best possible, we will make o eort to d the best possible c ad. We will prove that for 0 ; (2) holds with c = 1 10 ad = 1 60. For sucietly large, choose m of the form m = 2 t?1 where t 3 is a iteger, so that The rst author is partly supported by the NSF grat DMS 9704114 ad the grat MR1-181 of the Cooperative Grat Program of the Civilia Research ad Developmet Foudatio. The secod author is partly supported by the NSF grat DMS 9801396.
log 2 2 log 2 m 4 Costructio of G: Cosider a biary tree B with 1 + 2 + + 2 t?1 vertices ad L(B) be the set of all leafs. Let B 1 ad B 2 be two disjoit copies of B with roots x 1 ad x 2 respectively. Let T be a tree deed by ad V (T ) = V (B 1 ) [ V (B 2 ) [ fy 1 ; y 2 g E(T ) = E(B 1 ) [ E(B 2 ) [ fx 1 ; y 1 g [ fy 1 ; y 2 g [ fx 2 ; y 2 g That is, T is a biary tree with 2 t+1 vertices ad rooted edge fy 1 ; y 2 g. Let L(T ) = L(B 1 ) [ L(B 2 ) be a set of all leafs of T. Fially, cosider a cycle H ' C 2m of legth 2m ad the vertex set V (H) = L(T ). Cosiderig labeled vertices of L(T ) there are clearly (2m)! 2m of such cycles H. Label them Hi for i = 1; 2; :::; (2m)! 2m. For each H i, let TH i be a copy of T so that TH i 's are vertex disjoit for distict i's. For each i (2m)! 2m we also set H ~ i = Hi[TH i. Let H be the set of all graphs H ~ i costructed above. Note that for each H ~ i 2 H ay automorphism ' of H ~ i satises '(E(TH i )) = E(TH i ). This is due to the fact that the oly vertices of degree two i H ~ i are y 1 ad y 2, so they must be mapped to fy 1 ; y 2 g. So, x 1 ad x 2 must be mapped to fx 1 ; x 2 g sice they ca't be mapped to fy 1 ; y 2 g ad if they were mapped aywhere else the they would't be adjacet to '(y 1 ) ad '(y 2 ) respectively. Similarly a vertex o ay level of the tree must be mapped by ' to a vertex o that same level. Thus, i particular, ay automorphism of H ~ i is a automorphism of TH i. O the other had, there are 2 1 2 2 2 22 : : : 2 2t?1 < 2 2t = 2 2m automorphisms of TH i ' TH ad hece there are at least (2m)! 2m? 2 2m = (2m? 1)! 2 2m > m m distict automorphism types amog the graphs ~ H i 2 H. Observe that m m > log2 2 = 2 2 (1? ) > > q = b c 4m ad without loss of geerality assume that f ~ H1 ; ~ H2 ; : : : ~ H qg are pairwise oisomorphic. We dee G to be a disjoit uio of ~ H i; 1 i q. Note that G has at most vertices ad maximum degree 3 while the miimum degree is two. This meas that (3) (G) 3 5 :
Set l = 1 1 10 2( 15m log ) 2 = 1 10 15m 1 ad observe that 1 15m 2 60 log 2 ( ) 1 60 Note that Theorem 1 immediately follows from the followig. Fact. If F is a graph of l edges, the F 6?! G. Proof of Fact: P Set k = 10l: Let Vhigh = fv 2 V (F ); deg(v) > kg: Sice 2l fdeg(v); v 2 Vhighg we ifer that jvhighj < 5 Now we focus o vertices ad edges spaed by the set Vlow = V? Vhigh: For a set X Vlow let F [X] be a subgraph of F iduced by X. We say that a edge e Vlow ca see a 2m elemet set S Vlow if there exists a set R Vlow; jrj = 2(1 + 2 + : : : 2 t?2 ) ad edge preservig 1? 1 mappig (isomorphism ito) ' : T?! F [e[r [ S] such that '(fy 1 ; y 2 g) = e ad '(L(T )) = S If moreover H is a cycle of legth 2m; V (H) Vlow ad for some i = 1; 2; : : : q, ' is a 1? 1, edge preservig mappig ' : ~ H i?! F [e [ R [ V (H)] such that '(fy 1 ; y 2 g) = e ad '(L(T )) = V (H) we will say that e ca see H by i. V low e H i V high Figure 1. The edge e ca see H by i. Fix a arbitrary edge e Vlow. Due to the fact that deg(v) k for each v 2 Vlow, there are at most k 2 k 2 2(1+2+:::+2 t?2 ) k 8m
sets of size 2m that ca be see from e. Sice there are o more tha k 2m cycles C 2m spaed o ay 2m elemet set S, we ifer that edge e ca see by some i at most k 8m k 2m = k 10m cycles C 2m. Set Elow = E(F [Vlow]) ad cosider a auxilliary bipartite graph? with vertex set V (?) = Elow [ f1; 2; : : : ; qg For e 2 Elow ad i 2 f1; 2; : : : qg we set fe; ig 2 E(?) if there exists a cycle of legth 2m, V (H) Vlow so that e ca see H by i. Sice deg? (e) k 10m ad jelowj l, there exists i 0 with deg? (i 0 ) lk10m 5mlk 10m log 20 2 q log ( 2 ) 10m 15m = o(): I other words, there exists i 0 2 f1; 2; : : : qg so that the set N? (i) of?-eighbours of i 0 (i.e. edges which ca see a cycle H ' C 2m by i 0 ) has cardiality o(). Colorig F. We color N? (i 0 ) together with all edges icidet to Vhigh red. All remaiig edges are colored blue. Let F = F red [F blue be a colorig described above ad let (F red ) = M ifjw j : W V (F red ) ad e \ W 6= 0 for all e 2 E(F red )g. Observe that (F red ) + o() 5 < 2 while (G), the miimum umber of vertices of G which are icidet 5 to all edges of G satises (G) = jv (G)j? (G) =? (G) 2. Cosequetly, 5 (F red ) < (G). Suppose that there is a red copy G red of G. This would, however, mea that (G red ) (F red ) which is a cotradictio. O the other had, sice all edges that ca see a cycle of legth 2m by i 0 are colored red, H ~ i 0 ad cosequetly G are ot subgraphs of F blue. Cocludig Remark: Set ^r(; ) = M axg^r(g), where the maximum is take over all graphs G with vertices ad maximum degree. We cojecture that for ay 3 there is > 0 such that 1+ ^r(; ) 2? Ackowledgemet: We thak Jaso Hut for may commets which improved the mauscript. Refereces [B1] J. Beck, O Size Ramsey Number of Paths, Trees ad Circuits I, J. Graph Theory 7, 115-129, 1983. [B2] J. Beck, O Size Ramsey Number of Paths, Trees ad Circuits II, I: Mathematics of Ramsey Theory (ed. J. Nesetril ad V. Rodl), 34-45, Spriger-Verlag, 1990. [E] P. Erd}os, O the combiatorial problems which I would most like to see solved, Combiatorica 1, 25-42, 1981. [EFRS] P. Erd}os, R. J. Faudree, C. C. Rousseau ad R. H. Schelp, The size Ramsey umber, Period, Math. Hug. 9, 145-161, 1978.
[FP] J. Friedma ad N. Pippeger, Expadig graphs cotai all small trees, Combiatorica 7, 71-76, 1981. [HKL] P. E. Haxell, Y. Kohayahawa ad T. Luczak, The iduced size-ramsey umber of cycles, Combi. Probab. Comput. 4, 217-239, 1995. Emory Uiversity, Atlata, GA 30322, USA, rodl@mathcs.emory.edu Rutgers Uiversity, Piscataway, NJ 08854, USA, szemered@cs.rutgers.edu