Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γ c (G) + 2, where γ c (G) is the connected domination number of G. Bounds of the form diameter(G) ≤ rc(G) ≤ diameter(G) + c, 1 ≤ c ≤ 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) ≤ 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds.An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from [Schiermeyer, 2009], improving the previously best known bound of 20n/δ [Krivelevich and Yuster, 2010]. Moreover, this bound is seen to be tight up to additive factors by a construction mentioned in [Caro et al., 2008].

A sequence f:false[nfalse]→double-struckR contains a pattern π∈scriptSk, that is, a permutations of [k], iff there are indices i1 < … < ik, such that f(ix) > f(iy) whenever π(x) > π(y). Otherwise, f is π‐free. We study the property testing problem of distinguishing, for a fixed π, between π‐free sequences and the sequences which differ from any π‐free sequence in more than ϵ n places. Our main findings are as follows: (1) For monotone patterns, that is, π = (k,k − 1,…,1) and π = (1,2,…,k), there exists a nonadaptive one‐sided error ϵ‐test of false(ϵ−1normallognfalse)Ofalse(k2false) query complexity. For any other π, any nonadaptive one‐sided error test requires normalΩfalse(nfalse) queries. The latter lower‐bound is tight for π = (1,3,2). For specific π∈scriptSk it can be strengthened to Ω(n1 − 2/(k + 1)). The general case upper‐bound is O(ϵ−1/kn1 − 1/k). (2) For adaptive testing the situation is quite different. In particular, for any π∈scriptS3 there exists an adaptive ϵ‐tester of false(ϵ−1normallog1emnfalse)Ofalse(1false) query complexity.

In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → R of length n contains a pattern π ∈ S k (S k is the group of permutations of k elements), iff there are indices i 1 < i 2 < · · · < i k , such that f (i x ) > f (i y ) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ S k , k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than n places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error -tests of (complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error -test requires at least Ω( √ n) queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n 1−2/(k+1) ).On the other hand, there always exists a nonadaptive one-sided error -test for π ∈ S k with O( −1/k n 1−1/k ) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = * Department of Computer Science, University of Haifa, Israel. (1, 3, 2), we describe an -test with (almost tight) query complexity of O( √ n).Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ S 3 , tests π-freeness by making ( −1 log n) O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤ r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K 1,n for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤ rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 − 1, where n is order of the graph [1].It is known that computing rc(G) is NP-Hard [2]. Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

Abstract. A family F of permutations of the vertices of a hypergraph H is called pairwise suitable for H if, for every pair of disjoint edges in H, there exists a permutation in F in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for H is called the separation dimension of H and is denoted by π(H). Equivalently, π(H) is the smallest natural number k so that the vertices of H can be embedded in R k such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph H is equal to the boxicity of the line graph of H. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.