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].

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An axis-parallel b-dimensional box is a Cartesian product R 1 × R 2 × · · · × R b where R i (for 1 i b) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b such that G is representable as the intersection graph of (axis-parallel) boxes in b-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research, etc. Though many authors have investigated this concept, not much is known about the boxicity of many wellknown graph classes (except for a couple of cases) perhaps due to lack of effective approaches. Also, little is known about the structure imposed on a graph by its high boxicity.The concepts of tree decomposition and treewidth play a very important role in modern graph theory and have many applications to computer science. In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, box(G) tw(G) + 2, where box(G) and tw(G) denote the boxicity and treewidth of G, respectively. We also show that this upper bound is (almost) tight. Since treewidth and tree decompositions are extensively studied concepts, our result leads to various interesting consequences, like bounding the boxicity of many well-known graph classes, such as chordal graphs, circular arc graphs, AT-free graphs, co-comparability graphs, etc. For all these graph classes, no bounds on their boxicity were known previously. All our bounds are shown to be tight up to small constant factors. An algorithmic consequence of our result is a linear time algorithm to construct a box representation for graphs of bounded treewidth in a constant dimensional space.

A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2D 2 . We also conjecture that the best upper bound is linear in D.

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a (G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.