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 k-dimensional box is the cartesian product R 1 × R 2 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R 1 × R 2 × · · · × R k where each R i is a closed interval on the real line of the form [a i , a i + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) ≤ t + ⌈log (n − t)⌉ − 1 and box(G) ≤ t 2 + 1, where t is the cardinality of the minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds.F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, box(G) ≤ n 2 , where n is the number of vertices of G, and this bound is tight. We show that if G is a bipartite graph then box(G) ≤ n 4 and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to n 4 . Interestingly, if boxicity is very close to n 2 , then chromatic number also has to be very high. In particular, we show that if box(G) = n 2 −s, s ≥ 0, then χ(G) ≥ n 2s+2 , where χ(G) is the chromatic number of G.

Abstract:The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree , the rainbow connection number is upper bounded by 3n /( +1)+3. As an intermediate step we obtain an upper bound of 3n /( +1)−2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree . This bound is tight up to an additive constant of 2. This result may be of independent interest. Journal of Graph Theory

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A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (src(G)) of the graph. When proving upper bounds on src(G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number (vsrc(G)). In this paper, we give upper bounds on vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on src(G) for these classes, showing that the study of vsrc(G) enables meaningful progress on bounding src(G). Then we study the complexity of the problem to compute vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for src(G). We also observe that deciding whether vsrc(G) = k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether vsrc(G) ≤ 3 nor to approximate vsrc(G) within a factor n 1−ε , unless P=NP.