ONLINE VERTEX COLORINGS OF RANDOM GRAPHS WITHOUT MONOCHROMATIC SUBGRAPHS


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1 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS WITHOUT MONOCHROMATIC SUBGRAPHS MARTIN MARCINISZYN AND RETO SPÖHEL Abstract. Consider the following generalized notion of graph colorings: a vertex coloring of graph G is valid w.r.t. some fixed nonempty graph F if no color class induces a copy of F in G, i.e., there is no monochromatic copy of F in G. We propose and analyze an algorithm for computing valid colorings of a random graph G n,p on n vertices with edge probability p in an online fashion. For a large family of graphs F including cliques and cycles of arbitrary size, the proposed algorithm is optimal in the following sense: for any integer r 1, there is a constant β = β(f, r) such that the algorithm a.a.s. (asymptotically almost surely) computes a valid rcoloring of G n,p w.r.t. F online if p n β, and any online algorithm will a.a.s. fail to do so if p n β. That is, we observe a threshold phenomenon determined by the function n β. 1. Introduction The chromatic number χ(g) of a graph G is the minimum number of colors r such that there exists an rvertexcoloring of G in which no edge has endpoints of the same color. Computing the chromatic number of a graph is a wellknown NPcomplete problem [8]. Moreover, very deep results [5, 6, 13] show that we cannot even efficiently approximate the value of χ(g) of graphs G on n vertices within a factor of n 1 ε for any constant ε > 0 unless NP ZP P. (ZP P is the class of languages decidable by a random expected polynomialtime algorithm that makes no errors.) Many bounds on the chromatic number can be found in the literature. Clearly, the size of the largest clique ω(g) in graph G is a lower bound on χ(g). On the other hand, (G) + 1 is an upper bound on the chromatic number, where (G) denotes the maximum degree in G as usual. This bound can be obtained from a simple first fit algorithm. Suppose G is a graph on the vertex set {v 1,..., v n }, and r := (G) + 1 colors are identified by numbers 1,..., r. First fit considers the vertices of G in the order v 1,..., v n. To each vertex v i, 1 i n, it assigns the smallest color available that does not create a monochromatic edge, i.e., the smallest color index k, 1 k r, such that no vertex v j adjacent to v i with j < i was assigned to k before. This obviously produces a valid rcoloring of G. Employing a more intelligent ordering of the vertices, the same algorithm yields the theorem of Brooks [3]. This classical result states that for every graph G, we have χ(g) (G) unless G contains a clique on (G) + 1 vertices or an odd cycle if (G) = 2. We are interested in onlinecolorings of random graphs. We consider the following graph process starting with the empty graph G 0. In each step i > 0, a new vertex v i appears and is randomly connected to the vertices of G i 1 by independently including each of the edges {v i, v j }, 1 j < i with probability p into a new graph G i. After n steps this process results in a random graph G = G n,p, in which each edge is present with probability p independently of all other edges. An online coloring strategy must assign a color to v i in each step i immediately, thereby extending the coloring of G i 1 to G i in a proper way. It is wellknown [2, 9, 11] that in random graphs with p n 1, first fit asymptotically almost surely (a.a.s., i.e., with probability tending to 1 as n goes to infinity) needs about 2χ(G n,p ) colors. Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland The second author was supported by SNF grant
2 2 MARTIN MARCINISZYN AND RETO SPÖHEL In sparser random graphs, first fit performs not that well. For p = c/n, χ(g n,p ) is a.a.s. bounded by a constant only depending on c. However, Pittel and Weishaar [16] showed that first fit a.a.s. uses about log 2 log n colors in random graphs of this density. We view the question from the opposite perspective: supposing that only a constant number of colors r is available, we ask for which values of p an online coloring strategy exists that a.a.s. produces a valid coloring of G n,p. In this paper, we primarily consider a more general notion of valid colorings than described above. For the classical case, our results yield that first fit a.a.s. colors G n,p with r colors as long as p n 1/(1 2 r), and that no online algorithm can do substantially better. The maximal degree in G n,p with p slightly below this bound is roughly 2 r, so first fit a.a.s. performs exponentially better than what is guaranteed by Brooks theorem. On the other hand, G = G n,p is a.a.s. a forest and has chromatic number χ(g) = 2 for p n 1. In this paper, we consider the following widely used generalized notion of vertex colorings of graphs. Rather than requiring that none of the color classes induces an edge in G, we call an r coloring valid w.r.t. a fixed nonempty graph F if no color class induces a copy of F in G, i.e., there is no monochromatic copy of F in G. Suppose r 1 is a fixed integer. We propose and analyze an algorithm for computing valid rcolorings of random graphs online. For notational reasons that will become clear later, we consider the r available colors in the reverse order r,..., 1. The simple greedy F avoidance strategy of selecting the largest color index that allows for a valid extension of the coloring turns out to be not optimal in general. For example, consider the graph F consisting of a triangle with one edge attached to it. Here our results yield that greedily avoiding triangles and forgetting about the additional edge is better than greedily avoiding the entire graph. Therefore, we consider the following refinement. Let H 1,..., H r be a sequence of fixed nonempty subgraphs H k F, 1 k r. Then the greedy H 1,..., H r avoidance strategy chooses, in each step i, the maximal color index k such that v i is not contained in a copy of H k induced by the color class k in G i. In other words, it avoids monochromatic copies of H k in color k, 1 k r. (The inclusion of H 1 may seem artificial but turns out to be convenient.) Clearly, if we succeed in assigning a color to each vertex of G in this way, we obtain a valid rcoloring w.r.t. the graph F. We show how to choose the subgraphs H 1,..., H r such that, for a large family of graphs F, the resulting greedy H 1,..., H r avoidance strategy is optimal in the following sense: there is a constant β = β(f, r) such that the greedy H 1,..., H r avoidance strategy a.a.s. computes a valid rcoloring w.r.t. F of G n,p if p n β, and any online strategy will a.a.s. fail to do so if p n β. That is, we observe a threshold phenomenon determined by the function n β. More precisely, the value of β is reciprocal to the following recursively defined density measure for graphs, where we denote E(H) by e H and V (H) by v H : m 0 1(F ) := 0 and m k 1(F ) := max H F e H + m k 1 1 (F ) v H k 1. (1) For any given graph F, we define the following infinite sequence of graphs accordingly: Our main result reads as follows. H k = H k (F ) := argmax H F e H + m k 1 1 (F ) v H k 1. (2) Theorem 1. Let F be a nonempty graph and r 1 an integer. Let H 1,..., H r F be chosen according to (2). Then the greedy H 1,..., H r avoidance strategy a.a.s. produces an rcoloring of G n,p without any monochromatic copy of F online provided we have p = p(n) n 1/mr 1 (F ).
3 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS 3 Moreover, assuming there exists an induced subgraph F F on v F 1 vertices that satisfies this strategy is optimal. e H max H F v H 1 m2 1(F ), (3) We refer to the greedy H 1,..., H r avoidance strategy with H 1,..., H r as in (2) as the enhanced first fit algorithm. Assumption (3) is inherent to our method for proving the optimality of this strategy. It holds for a large family of graphs including cliques and cycles of arbitrary size. In these cases, we also obtain a closed form for the function m r 1 (F ). As usual, let K l and C l denote a clique and a cycle on l vertices respectively. As those graphs have a highly regular structure, we have H k = F for all k 1, i.e., the simple first fit algorithm is optimal. Corollary 2 (Cliques). For all l 2 and r 1, the greedy K l avoidance strategy a.a.s. produces an rcoloring of G n,p without any monochromatic clique of size l online provided we have and this strategy is optimal. p = p(n) n 2 l(1 l r ), Corollary 3 (Cycles). For all l 2 and r 1, the greedy C l avoidance strategy a.a.s. produces an rcoloring of G n,p without any monochromatic cycle of length l online provided we have and this strategy is optimal. p = p(n) n l 1 l(1 l r ), 1.1. Bounds on the threshold function. Two classical results on random graphs yield obvious lower and upper bounds on the threshold for onlinecolorability with a fixed number of colors r. Clearly, if after the last step of the online process G = G n,p contains no copy of F, extending the coloring of any intermediate graph G i from i to i + 1 vertices is trivial. The wellknown theorem of Bollobàs [1], which is a generalization of a result of Erdős and Rényi [4] to arbitrary graphs F, states a threshold for this event. Theorem 4. Let F be a nonempty graph. Then we have lim P[F G n,p] = n { 0 if p n 1/m(F ) 1 if p n 1/m(F ), where e H m(f ) := max. H F v H Therefore, for p n 1/m(F ), there is a.a.s. no copy of F in G n,p, and any coloring algorithm will succeed. In fact, we have m 1 1 (F ) = m(f ), and setting r = 1 in Theorem 1 yields the statement of Theorem 4. It is easy to see that for all r 2, m r 1 (F ) is strictly greater than m(f ) and thus yields a higher threshold. An upper bound on the threshold is obtained by studying offline colorings of random graphs. Obviously, the existence of a valid rcoloring of G = G n,p is necessary for any strategy to succeed. A result of Luczak, Ruciński, and Voigt [12] states that, beyond a certain threshold for the edge probability p, G a.a.s. admits no rcoloring without a monochromatic copy of F. Theorem 5. Let r 2 and F be a nonempty graph that is not a matching if r = 2. Moreover, let G (F ) v r denote the property that every rcoloring of G contains a monochromatic copy of F.
4 4 MARTIN MARCINISZYN AND RETO SPÖHEL Then there exist positive constants c = c(f, r) and C = C(F, r) such that { lim P[G n,p (F ) v 0 if p cn 1/m 1(F ) n r] = 1 if p Cn 1/m, 1(F ) where m 1 (F ) := max H F e H v H 1. Hence, there is no hope of finding an rcoloring avoiding a monochromatic copy of F whenever p n 1/m 1(F ). Note that the function m 1 (F ) determining the order of magnitude of the threshold does not depend on the number of available colors r in the offline case. In the online case, however, the density function m r 1 (F ) is monotone increasing with the number of available colors r. It is easily deduced from the formulas in Corollaries 2 and 3 that, in the case of cliques and cycles, the online threshold converges to the offline threshold as r tends to infinity. One can show that this convergence holds in general, i.e., that all graphs F satisfy lim r mr 1(F ) = m 1 (F ). It follows that for any constant ε > 0, the enhanced first fit algorithm a.a.s. produces a valid rvertexcoloring of G n,p with p n 1/m 1(F ) ε if r is sufficiently large Related Work. The problem of finding thresholds for onlinecolorability with a fixed number of colors was first considered for edgecolorings of random graphs. Investigating algorithmic Ramsey properties of triangles, Friedgut et al. [7] introduced and studied the online edge coloring game. In this game we start with an empty graphs on n vertices and new edges arrive in a random order. A single player must assign one of r available colors to each new edge immediately such that no monochromatic copy of a given graph F is created. The game ends as soon as the player is forced to close such a monochromatic copy. The authors of [7] showed that there exists a threshold for the duration of the game in the case when F is a triangle and r = 2. They proved that the player can a.a.s. avoid monochromatic triangles for any N n 4/3 random edges by playing greedily. If, however, N n 4/3 random edges are inserted, the player a.a.s. loses the game no matter which strategy is chosen. This result was generalized in [14] and [15] to a large family of graphs F including cliques and cycles of arbitrary size. It was shown that for those graphs F, the threshold for the duration of the F avoidance edge coloring game is given by where N 0 (n) := n 2 1/m 2(F ), m 2 (F ) := max H F e H v H 2 + 1/m(F ). It seems plausible that m 2 (F ) determines a threshold for all graphs F other than forests, and that the theory generalizes to the game with more than two colors as in the vertexcoloring case. However, for r 3, there are only nontrivial lower bounds on the duration of the F avoidance edge coloring game. 2. Preliminaries and notation The expression f g means that the functions f and g differ at most by a multiplicative constant. All graphs are simple and undirected. The number of vertices of a graph G is denoted by v G or v(g), and similarly the number of edges by e G or e(g).
5 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS Density measures for graphs. A density measure is a function mapping graphs to nonnegative reals. For simplicity, we define that all density measures evaluate to zero on empty graphs. The most wellknown density measure is d(h) := e H v H for any nonempty graph H. This is exactly the average degree of H divided by 2. The density measure d 1 (H) := e H v H 1 is also wellknown and motivated by Theorem 5. For any given density measure d i, we define the maximum density m i (H) := max d i(j) J H (cf. Theorems 4 and 5). We say that H is balanced with respect to d i if m i (H) = d i (H). H is balanced in the ordinary sense or simply balanced if it is balanced w.r.t. the density measure d. We call H 1balanced if it is balanced w.r.t. the density measure d 1. It is wellknown that every graph H satisfies m(h) m 1 (H), and that every 1balanced graph is also balanced. Reflecting the maximum density measure m r 1 (H) as defined in (1), for any r 1 and nonempty graphs H and J, let d r 1(H, J) := e J + m r 1 1 (H) v J. We abbreviate d r 1(H, H) by d r 1(H). We say that H is balanced w.r.t. d r 1 if m r 1 (H) = dr 1(H). By definition, we have m 1 1 (H) = m(h) and d1 1(H) = d(h). We use the shorthand notation m 1 for m 2 1 and d 1 for d 2 1 respectively. The maximum density m r 1 was apparently not studied before in the literature. The next lemma shows that it is sandwiched between m and m 1. The proof involves only elementary calculations and is omitted in this extended abstract. Lemma 6. (i) Every nonempty graph F satisfies (ii) Every graph F satisfies m(f ) = m 1 1(F ) < m 2 1(F ) < < m r 1(F ) < < m 1 (F ). lim r mr 1(F ) = m 1 (F ). (iii) Every 1balanced graph is balanced with respect to d r 1 for all r 1. (iv) For r 2, every graph that is balanced with respect to d r 1 is balanced with respect to d r 1 1. It is easily verified that the inductive definition of m r 1 (F ) can be written in the following explicit form, which yields the threshold formulas in Corollaries 2 and 3. The proof proceeds by induction on r. Lemma 7. For all nonempty graphs F and r 1, we have If F is 1balanced, we have m r 1(F ) = max H 1,...,H r F m r 1(F ) = r i=1 e(h i) i 1 j=1 v(h j) r i=1 v(h i) e F v F 1 ( ) 1 v r F.. (4)
6 6 MARTIN MARCINISZYN AND RETO SPÖHEL 2.2. Janson s inequality. Janson s inequality [10] is a very useful tool in probabilistic combinatorics. In many cases, it yields an exponential bound on lower tails where the second moment method only gives a bound of o(1). Here we formulate a version tailored to random graphs. Theorem 8. Consider a family F of graphs on the vertex set [n]. For each H F, let X H denote the indicator random variable for the event H G n,p, and for each pair H 1, H 2 F, H 1 H 2, write H 1 H 2 if H 1 and H 2 are not edgedisjoint. Let X = H F X H, µ = E[X] = H F p e(h), = H 1,H 2 F H 1 H 2 E[X H1 X H2 ] = H 1,H 2 F H 1 H 2 e(h p 1 )+e(h 2 ) e(h 1 H 2 ). Then, for all 0 δ 1, we have P[X (1 δ)µ] e δ 2 µ 2 2(µ+ ). 3. Analysis of the enhanced first fit algorithm In this section we analyze the enhanced first fit algorithm as proposed in the introduction and show that it a.a.s. produces valid rcolorings of G n,p w.r.t. F up to the edge probability p 0 = p 0 (F, r, n) := n 1/mr 1 (F ). We present the proofs from a game perspective as in Section 1.2. A single player called Painter is given the vertices of a random graph one by one in the manner described in the introduction, and has to color them immediately. Her aim is to color the entire graph without creating a monochromatic copy of some fixed graph F. Consider the game with two colors, say red (2) and blue (1), and suppose Painter plays the greedy H 1, H 2 avoidance strategy with H 1, H 2 F fixed. That is, she uses red in every step unless this creates a red copy of H 2. Note that choosing H 1 F implies that the game comes to a premature end as soon as a blue copy of H 1 is created. As will become evident soon, this subtle technicality makes no substantial difference provided H 1 is suitably chosen, but greatly improves readability. Thus, when Painter loses the game, the random graph contains an entirely blue copy of H 1 with the following property: recoloring any vertex of this copy of H 1 to red creates an entirely red copy of H 2. We say that every vertex of this copy of H 1 closes a red copy of H 2. This motivates the definition of the following family of graphs. Let F(H 1, H 2 ) denote the class of all graphs that have an inner (blue) copy of H 1, each vertex of which closes an outer (red) copy of H 2. Here the colors should only provide the intuitive connection to the greedy strategy, the members of the family F(H 1, H 2 ) are not associated with a coloring. Two graphs from the family F(K 4, K 4 ) are depicted in Figure 1. We say that the inner copy of H 1 is formed by inner vertices and edges, and refer to the surrounding elements as outer vertices and edges respectively. Observe that in this terminology every outer copy contains exactly one inner vertex and no inner edges. We formally define this family of graphs as follows.
7 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS 7 Figure 1. Two graphs from the family F(K 4, K 4 ). The graph on the left hand side is the unique member with vertexdisjoint outer copies. In both graphs, the inner vertices and edges are shaded. Definition 9. For all graphs H 1 = (V, E) and H 2, let { F(H 1, H 2 ) := F = (V. U, E. D) : F is a minimal graph such that for all v V there are sets U(v) U and D(v) D with ( v. ) } U(v), D(v) = H2 We take F as a minimal element with respect to subgraph inclusion, i.e., F does not have a subgraph which satisfies the same properties. This ensures in particular that F(H 1, H 2 ) is finite. In the game with three colors, say, with colors yellow (3), red (2) and blue (1), a greedy Painter first avoids some subgraph H 3 in yellow, H 2 in red, and eventually H 1 in blue. By the same argument as in the case of two colors, when Painter loses the game, the random graph contains a redblue copy of a member from the family F(H 1, H 2 ), every vertex of which closes an entirely yellow copy of H 3. We denote the class of all such graphs by F(H 1, H 2, H 3 ). This motivates the following inductive definition for general r. For notational reasons, we include the game with one color into the formalism. Definition 10. For any graph H 1, let F(H 1 ) := {H 1 }. For r 2 and arbitrary graphs H 1,..., H r, let F(H 1,..., H r ) := { F F(F, H r ) : F F(H 1,..., H r 1 ) }. With these definitions at hand, we can state the following simple observation. Lemma 11. Let F be a nonempty graph and r 1 an integer. Let H 1,..., H r F be arbitrary nonempty subgraphs. Then Painter a.a.s. wins the online vertex coloring game w.r.t. F with r colors if playing the greedy H 1,..., H r avoidance strategy as long as p n 1/g(H 1,H 2,...,H r), where g(h 1, H 2,..., H r ) := min { m(f ) : F F(H 1, H 2,..., H r ) }. Proof. As argued before, a member of the family F(H 1,..., H r ) must be contained in G n,p if Painter loses greedily avoiding H 1,..., H r. By Theorem 4 and the definition of g, such a graph does a.a.s. not exist in G n,p with p n 1/g(H 1,...,H r). Now, the first part of Theorem 1 follows from Lemma 11 and the next lemma. Lemma 12. Let F be a nonempty graph and r 1 an integer. If the sequence of subgraphs H 1,..., H r is chosen according to (2), all graphs F F(H 1,..., H r ) satisfy m(f ) m r 1(F )..
8 8 MARTIN MARCINISZYN AND RETO SPÖHEL Lemmas 11 and 12 imply together that Painter can a.a.s. successfully color the entire graph G n,p without any monochromatic copy of F online as long as p n 1/mr 1 (F ). It remains to prove Lemma 12. In this extended abstract, we only outline the underlying ideas of this proof. For the sake of simplicity, consider the case r = 2. Among the members of the family F(H 1, H 2 ), there are some distinguished ones that have vertexdisjoint outer copies of H 2. The left hand side of Figure 1 shows an example for the case H 1 = H 2 = K 4. Consider any such member F 2 F(H 1, H 2 ) with vertexdisjoint outer copies. Clearly, we have due to the choice of H 1 and H 2. graphs F F(H 1, H 2 ), we have e(f 2 ) v(f 2 ) = v H 1 e H2 + e H1 v H1 v H2 = e H 2 + e H1 /v H1 v H2 = m 1 (F ) Hence, Lemma 12 is equivalent to the assertion that for all m(f ) e(f 2 ) v(f2 ). Thus, the statement of Lemma 12 is essentially that members of the family F(H 1, H 2 ) that contain overlapping substructures are at least as dense as the nice members F2. A formal proof is not hard to obtain, but slightly technical. It is omitted in this extended abstract. For r 3, the lemma follows by induction. Note that the right hand side of (4) is exactly the density of a nice graph Fr F(H 1,..., H r ) in which all outer copies are vertexdisjoint. 4. Optimality of the enhanced first fit algorithm In this section, we prove that the enhanced first fit algorithm is optimal in the sense that any online algorithm a.a.s. fails to produce a valid rcoloring of G n,p w.r.t. F if p n 1/mr 1 (F ). We employ a multiround exposure approach, in each round using the following counting version of Theorem 5. The proof of Theorem 13 proceeds along the same lines as the upper bound proof of Theorem 5, utilizing Janson s inequality. It is omitted due to space restrictions. Theorem 13. Let r 1 and F be a nonempty graph. Then there exist constants C = C(F, r) > 0 and a = a(f, r) > 0 such that for all p(n) Cn 1/m 1(F ), a.a.s. in every rvertexcoloring of G n,p, one color class induces at least an v F p e F monochromatic copies of F. The assumption that there exists an induced subgraph F F on v F 1 vertices satisfying m 1 (F ) m 1 (F ) (cf. (3)) guarantees that we can apply Theorem 13 to F. Rather than generating the random graph G n,p online as stated in the introduction, it is more convenient here to generate the entire graph in advance. The random experiment is conducted for all edges before the game starts. In each move i, one new vertex v i is revealed to Painter together with all edges connecting v i to the vertices v j, 1 j < i. Edges that will appear in the future are hidden from Painter s view. Assume that p n 1/mr 1 (F ) is given. We prove by induction on r that Painter a.a.s. loses the game regardless of the strategy she employs. The case r = 1 follows directly from Theorem 4. Suppose we have r 2 and the statement holds for r 1. We partition the vertex set of G = G n,p into two sets V 1 := {v 1,..., v n/2 } and V 2 = {v n/2 +1,..., v n } and relax the game to a semionline game, revealing the subgraph G n/2 = G[V 1 ] to Painter all at once. She may color the vertices of V 1 offline. In the second round, the vertices of V 2 are revealed one by one, and Painter has to color them online as before.
9 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS 9 Suppose that Painter s coloring of the vertex set V 1 is fixed and consider the set of edges E G (V 1, V 2 ) generated between V 1 and V 2, but hidden from Painter s view. For each color k {1,..., r}, this edge set defines a vertex set Base(k) V 2 consisting of all vertices in V 2 that complete a copy F G[V 1 ] in color k to a copy of F. Obviously, Painter may not assign color k to any vertex in Base(k) in the second round. The next claim asserts that there exists some color class k 0 {1,..., r} such that Base(k 0 ) is sufficiently large so that we can apply the induction hypothesis. Claim 14. After the first round was played, there a.a.s. exists a color k 0 {1,..., r} such that we have Base(k 0 ) = Ω (n mr 1 1 (F )/m r 1 (F )). Claim 14 implies that in the second round, Painter must a.a.s. color a binomial random graph G en,ep on ñ n mr 1 1 (F )/m r 1 (F ) vertices with edge probability p = p n 1/mr 1 (F ) ñ 1/mr 1 1 (F ) with just r 1 colors left in an online fashion. Applying the induction hypothesis yields that she a.a.s. fails no matter which strategy she employs. Hence, it remains to prove Claim 14. Proof of Claim 14. In order to simplify notation, let β := m r 1 1 (F )/m r 1 (F ). Note that we have β < 1 due to Lemma 6. For any graph J F with v J 1 and p p 0 := n 1/mr 1 (F ), we have n v J p e J n v J e J /m r 1 (F ) n v J = n v J e J +v J m r 1 1 (F ) e J v J e J +m r 1 1 (F ) = n e J v J e J +m r 1 1 (F ) v J e J +m r 1 1 (F ) mr 1 1 (F ) n β, (5) with equality if p = p 0 and d r 1(F, J) = m r 1 (F ). By Theorem 13 and owing to m 1 (F ) m 1 (F ) m r 1 (F ) (cf. (3)), Painter s offlinecoloring of the vertices in V 1 a.a.s. creates Ω ( n v(f ) p e(f ) ) monochromatic copies of F. By the first moment method, there are a.a.s. at most O ( n v(f ) p e(f ) ) copies of F in total, and therefore there are a.a.s. Θ ( n v(f ) p e(f ) ) monochromatic copies of F in G[V 1 ]. W.l.o.g. we assume that those copies have color r. Suppose F 2 is a graph consisting of the union of two copies of F intersecting on some nonempty graph J F. Then there are a.a.s. at most O ( n 2v(F ) v(j ) p 2e(F ) e(j ) ) (6) copies of F 2 in G[V 1 ] by the first moment method. Since there are only constantly many nonisomorphic graphs F 2 with this structure, the bound in (6) a.a.s. hold for all of them in G[V 1 ]. Next, we estimate the probability that a fixed vertex v V 2 is contained in Base(r). Let u denote the missing vertex in F, i.e., {u} = V (F ) \ V (F ). For any subgraph J F, let Ĵ F denote the graph in which all edges leading from J to u are added, and let deg bj (u) = e(ĵ) e(j ) denote the degree of u in Ĵ. For any vertex v V 2 and any monochromatic copy Fi of F in G[V 1 ], let F v,i be the family of all minimal subsets of V 1 V 2 that connect Fi and v such that those form a copy of F. Note that F v,i has constant size depending only on the structure of F and F. If F is a clique, e.g., F v,i contains at most one member. Let F v := F i V 1 F v,i
10 10 MARTIN MARCINISZYN AND RETO SPÖHEL be a multiset allowing for duplicate entries since the same set of edges may complement distinct monochromatic copies of F in G[V 1 ] to F. For each member H F v, let X H denote the indicator random variable for the event H G n,p. Let X v := H F v X H. Clearly, the vertex v is contained in Base(r) if X v 1. We apply Theorem 8 to the family F v in order to obtain a lower bound on the probability of this event. Simple calculations yield that we have ( µ = E[X v ] = Θ n v(f ) ) p e(f ) p deg F (u) = Θ ( n v F 1 p e ) ( (5) F =Ω n β 1). Moreover, a.a.s. we have = H 1,H 2 F v E[X X ] H1 H2 H 1 H 2 Θ(1)p 2 deg F (u) deg J b(u) J F F e(j i,f j ) 1 Fi F j =J (6) = J F e(j ) 1 O ( n 2v(F ) v(j ) p 2e(F ) e(j ) ) p 2 deg F (u) deg b J (u) O ( n 2v F v J 1 p 2e ) F e J J F v J 1 = µ 2 J F v J 1 O ( n 1 v J p e ) ( (5) J =O µ 2 n 1 β). Therefore, Theorem 8 yields that { µ 2 } P[v / Base(r)] = P [X v = 0] exp 2(µ + ) { ( = exp Ω n β 1)} ( = 1 Ω n β 1). Since for two distinct vertices u, v V 2, the random variables X u and X v are independent, standard Chernoff bounds imply that a.a.s. we have Base(r) = Ω ( P [X v 1] = Ω n β). v V2 Note that this proof exhausts our approach for proving the optimality of the enhanced first fit strategy even for the game with just two colors: if (3) is violated, Painter can a.a.s. color all vertices of V 1 without creating a single monochromatic copy of F, and Base(1), Base(2) V 2 are both empty. Thus, there is little hope of proving general optimality of our strategy by considering the relaxation to a multiround offline game as proposed in this paper. References [1] B. Bollobás. Threshold functions for small subgraphs. Math. Proc. Cambridge Philos. Soc., 90(2): , [2] B. Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49 55, [3] R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philos. Soc., 37: , 1941.
11 ONLINE VERTEX COLORINGS OF RANDOM GRAPHS 11 [4] P. Erdős and A. Rényi. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 5:17 61, [5] U. Feige and J. Kilian. Zero knowledge and the chromatic number. J. Comput. System Sci., 57(2): , Complexity 96 The Eleventh Annual IEEE Conference on Computational Complexity (Philadelphia, PA). [6] M. Fürer. Improved hardness results for approximating the chromatic number. In Proceedings of the ThirtySixth Annual Symposium on Foundations of Computer Science, pages IEEE, [7] E. Friedgut, Y. Kohayakawa, V. Rödl, A. Ruciński, and P. Tetali. Ramsey games against a onearmed bandit. Combin. Probab. Comput., 12(56): , Special issue on Ramsey theory. [8] M. R. Garey and D. S. Johnson. Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., A guide to the theory of NPcompleteness, A Series of Books in the Mathematical Sciences. [9] G. R. Grimmett and C. J. H. McDiarmid. On colouring random graphs. Math. Proc. Cambridge Philos. Soc., 77: , [10] S. Janson. Poisson approximation for large deviations. Random Structures Algorithms, 1(2): , [11] T. Luczak. The chromatic number of random graphs. Combinatorica, 11(1):45 54, [12] T. Luczak, A. Ruciński, and B. Voigt. Ramsey properties of random graphs. J. Combin. Theory Ser. B, 56(1):55 68, [13] C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. J. Assoc. Comput. Mach., 41(5): , [14] M. Marciniszyn, R. Spöhel, and A. Steger. The online clique avoidance game on random graphs. In Approximation, randomization and combinatorial optimization, volume 3624 of Lecture Notes in Comput. Sci., pages Springer, Berlin, [15] M. Marciniszyn, R. Spöhel, and A. Steger. Online Ramsey games in random graphs. Submitted, [16] B. Pittel and R. S. Weishaar. Online coloring of sparse random graphs and random trees. J. Algorithms, 23(1): , 1997.
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