Rainbow saturation and graph capacities

Transcription

1 Rainbow sauraion and graph capaciies Dániel Korándi Absrac The -colored rainbow sauraion number rsa (n, F ) is he minimum size of a -edge-colored graph on n verices ha conains no rainbow copy of F, bu he addiion of any missing edge in any color creaes such a rainbow copy. Barrus, Ferrara, Vandenbussche and Wenger conjecured ha rsa (n, K s ) = Θ(n log n) for every s 3 and ( s. In his shor noe we prove he conjecure in a srong sense, asympoically deermining he rainbow sauraion number for riangles. Our lower bound is probabilisic in spiri, he upper bound is based on he Shannon capaciy of a cerain family of cliques. Inroducion A graph G is called F -sauraed if i is a maximal F -free graph. The classic sauraion problem, firs sudied by Zykov [4] and Erdős, Hajnal and Moon [4], asks for he minimum number of edges in an F -sauraed graph (as opposed o he Turán problem, which asks for he maximum number of edges in such a graph). A rainbow analog of his problem was recenly inroduced by Barrus, Ferrara, Vandenbussche and Wenger [], where a -edge-colored graph is defined o be rainbow F -sauraed if i conains no rainbow copy of F (i.e., a copy of F where all edges have differen colors), bu he addiion of any missing edge in any color creaes such a rainbow copy. Then he -colored rainbow sauraion number rsa (n, F ) is he minimum size of a -edge-colored rainbow F -sauraed graph. ( n log n log log n ) Among oher resuls, Barrus e al. showed ha Ω rsa (n, K s ) O(n log n) and conjecured ha heir upper bound is of he righ order of magniude: Conjecure. ([]). For s 3 and ( s, rsa (n, K s ) = Θ(n log n). Here we prove his conjecure in a srong sense: we give a lower bound ha is asympoically igh for riangles. Theorem.2. For s 3 and ( s, we have wih equaliy for s = 3. rsa (n, K s ) ( + o()) ( s + log( s + n log n We should poin ou ha Conjecure. was independenly verified by Girão, Lewis and Popielarz [9] and by Ferrara e al. [5], bu wih somewha weaker bounds. In fac, our resul proves a conjecure in [9], esablishing he sronger esimae rsa (n, K s ) = Θ s ( n log n log ) wih heir upper bound. Insiue of Mahemaics, EPFL, Lausanne, Swizerland. Research suppored in par by SNSF grans and daniel.korandi@epfl.ch.

2 Our lower bound is probabilisic in spiri, using ideas of Kaona and Szemerédi [0], and Füredi, Horak, Pareek and Zhu [6] (similar echniques were used in [2, 2, ]). The upper bound for s = 3 is based on he following heorem ha follows from a srong informaion-heoreic resul of Gargano, Körner and Vaccaro [8] on he Shannon capaciies of graph families. Theorem.3. For every 3, here is a se X [] k of m = ( ) ( o())k srings of lengh k from alphabe [] = {,..., } such ha for any x, x X and any a [], here is a posiion i where x(i) x (i) and x(i), x (i) a. In he nex secion we derive Theorem.3 from resuls abou he Shannon capaciy of graph families. This is followed by he proof of Theorem.2 in Secion 3. 2 Graph capaciies Le G = {G,..., G r } be a family of graphs on verex se []. Le N k be he maximum size of a se X [] k of srings of lengh k on alphabe [] such ha for any wo srings x, x X and any G j G, here is a posiion i j [k] such ha x(i j )x (i j ) is an edge in G j. The Shannon capaciy of he family G is defined as C(G) = lim sup k k log N k (see, e.g., [3, 3] ). When G = {G}, we simply wrie C(G) for C(G). We need an analogous definiion for srings where he occurrences of each a [] are proporional o some probabiliy measure P on []. So le T k (P, ε) be he se of all srings x [] k such ha k #{i : x(i) = a} P (a) < ε for every a [], and le M k,ε be he maximum size of a se X T k (P, ε) such ha for every x, x X here is an i wih x(i)x (i) G. The Shannon capaciy wihin ype P is C(G, P ) = lim ε 0 lim sup k k log M k,ε. Using a clever consrucion, Gargano, Körner and Vaccaro [8] showed ha C(G) can be expressed in erms of he C(G j, P ): Theorem 2. ([8]). For a family of graphs G = {G,..., G r } on verex se [], we have C(G) = max P min C(G j, P ). G j G In fac, hey proved a more general resul for Sperner capaciies, he analogous noion for direced graphs. Wha we need is a corollary ha follows easily from his heorem using sandard ools abou graph enropy (see he survey of Simonyi [3] for more informaion). Here we give a self-conained argumen ha goes along he lines of a proof by Gargano, Körner and Vaccaro [7] of he case s = 2. Corollary 2.2. Le 2 s be an ineger and le G be he family of all s-cliques on [] (each wih s isolaed verices). Then C(G) = s log s. Proof. For he lower bound, we can ake P o be he uniform measure on []. Then by Theorem 2., i is enough o show ha C(G, P ) s log s where G is a clique on [s] wih isolaed verices s +,...,. Le X k T k (P, k ) be he se of all srings x of lengh k such ha he firs sk/ The usual definiion is wih binary logarihm, bu he base of our logarihms is unimporan for our purposes. 2

3 leers of x conain k/ or k/ insances of each a [s], and x(i) = b for every s + b and < i bk. Then (b )k log(x k ) C(G, P ) lim = lim k k k k sk ( log )! (( k = lim )!)s k k log(ssk/ ) = s log s. For he upper bound, le X [] k be a maximum se of srings such ha for any x, x X and for every s-clique G G, here is an i [k] such ha x(i)x (i) G. We se m = X o be his maximum. We may assume ha {,..., s} are he s leas frequen elemens appearing in he srings of X. Le d x be he number of elemens in x X ha are no in [s], so x X d x s mk, and le X x be he se of srings obained from x by replacing hese elemens arbirarily wih numbers from [s]. Then X x = s dx, and X x, X x are disjoin for disinc x, x X because any sring from X x will differ from any sring in X x a he posiion i where x(i)x (i) is an edge of he clique on [s]. Then using Jensen s inequaliy we have s k x X s dx m s ( x X dx)/m m s ( s)k, and hence m s sk/, implying C(G) k log m s log s. Theorem.3 clearly follows from he case s =. 3 Rainbow sauraion Proof of Theorem.2. For he lower bound, suppose H is a -edge-colored rainbow K s -sauraed graph, and spli is verices ino wo pars: le A = {a,..., a k } be he se of verices of degree a leas d = log 3 n, and B be he res. We may assume A n log n (oherwise H has a leas 2 n log2 n edges), and hus B conains m ( log n )n verices. Now le us define a sring x v [ + ] k for every v B ha encodes he colors of he A-B edges ouching v as follows: x v (i) is + if a i v is no an edge in H, oherwise i is he color of a i v. Assume, wihou loss of generaliy, ha s + 3,..., are he s 2 mos common colors among he A-B edges. For v B, le X v [ s + 2] k be he se of srings obained from x v by replacing each s + 3,..., + wih an arbirary number from [ s + 2]. Then if d v denoes he number of A-B edges in H ouching v and d v denoes he number of such edges of colors s + 3,...,, hen X v = ( s + k dv+d v. We claim ha if v, w B are non-adjacen wih no common neighbor in B, hen X v and X w have no sring in common. Indeed, adding he edge vw of color creaes a rainbow K s wih s 2 verices in A. So here mus be an a i such ha a i v and a i w have differen colors, also differing from s + 3,...,. Bu hen all he srings in X v have he color of a i v as heir i h leer, and all he srings in X w have he color of a i w as heir i h leer, so X v and X w are disjoin. Since verices in B have degree a mos d, each v B has a mos d 2 verices w B ha are eiher adjacen o v or have a common neighbor wih v in B. So each sring in [ s + 2] k can appear 3

4 in no more han d 2 + collecions X w, and hence we ge (d 2 + )( s + k X v = s + v B v B( k dv+d v d 2 + ( s + d v d v m ( s + m ( v B d v v B dv) v B using Jensen s inequaliy. Now s + 3,..., were he s 2 mos common colors, so we also have v B d v s 2 and hus v B d v v B d v s 2 v B d v. Taking logs, we obain d v v B s + 2 m ( log s+2 m log s+2 (d 2 + ) ). v B d v As he lef-hand side is a lower bound on he number of edges in H, his esablishes he desired lower bound (using d = log 3 n and m = n + o(n)). For he upper bound in he case of riangles, le k be large enough, and ake a se X of size m as provided by Theorem.3. Consider a k-by-m complee biparie graph G 0 wih pars A and B, where A = {a,..., a k }, and B corresponds o he srings in X. For every verex v B, we look a he corresponding sring x X, and color each edge va i by he color x(i). G 0 is clearly (rainbow) riangle-free, and by he definiion of X, adding an edge o G 0 beween wo verices of B in any color a [] creaes a rainbow riangle. Now le G be a maximal rainbow riangle-free supergraph of G 0. Then G is rainbow rianglesauraed by definiion, and compared o G 0, i only has new edges induced by A, hus i has a mos km + ( ) k 2 edges. Here n = k + m and k = (+o()) log m, implying he required upper bound. ( ) log( ) For s > 3 our lower bound is probably no igh. asympoics of rsa (n, K s ) for general s. I would be ineresing o deermine he Acknowledgemens. I hank Shagnik Das for finding [7] for me, and Gábor Simonyi for some clarificaions abou capaciies. References [] M. D. Barrus, M. Ferrara, J. Vandenbussche and P. S. Wenger, Colored sauraion parameers for rainbow subgraphs, J. Graph Theory, 86 (207), [2] B. Bollobás and A. Sco, Separaing sysems and oriened graphs of diameer wo, J. Combin. Theory Ser. B 97 (2007), [3] I. Csiszár and J. Körner, Informaion Theory, 2nd ediion, Cambridge Universiy Press, 20. [4] P. Erdős, A. Hajnal and J.W. Moon, A problem in graph heory, Amer. Mah. Monhly, 7 (964),

5 [5] M. Ferrara, D. Johnson, S. Loeb, F. Pfender, A. Schule, H. C. Smih, E. Sullivan, M. Tai and C. Tompkins, On edge-colored sauraion problems, arxiv: preprin [6] Z. Füredi, P. Horak, C. M. Pareek and X. Zhu, Minimal oriened graphs of diameer 2, Graphs Combin. 4 (998), [7] L. Gargano, J. Körner and U. Vaccaro, Sperner capaciies, Graphs Combin., 9 (993), [8] L. Gargano, J. Körner and U. Vaccaro, Capaciies: from informaion heory o exremal se heory, J. Combin. Theory Ser. A, 68 (994), [9] A. Girão, D. Lewis and K. Popielarz, Rainbow sauraion of graphs, arxiv: preprin [0] G. Kaona and E. Szemerédi, On a problem of graph heory, Sudia Sci. Mah. Hungar. 2 (967), [] D. Korándi and B. Sudakov, Sauraion in random graphs, Random Srucures Algorihms 5 (207), [2] A. V. Kosochka, T. Luczak, G. Simonyi and E. Sopena, On he minimum number of edges giving maximum oriened chromaic number, in: Conemporary Trends in Discree Mahemaics, DIMACS Series in Discree Mahemaics and Theoreical Compuer Science, vol 49. (999), [3] G. Simonyi, Perfec graphs and graph enropy. An updaed survey, in: Perfec Graphs, Wiley (200), [4] A. Zykov, On some properies of linear complexes (in Russian), Ma. Sbornik N. S. 24 (949),