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Abstract: 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 such that there is an acyclic edge coloring using colors and it is denoted by . 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. [Copyright &y& Elsevier]

Abstract: In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, . The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph with a cost function is said to satisfy the -parametrized triangle inequality if for all distinct . Then the graph is called a -triangular graph. For any -triangular graph , for , we show that , where and are the costs of an optimum Hamiltonian cycle and an optimum cycle cover respectively. In addition, we observe that there exists an infinite family of -triangular graphs for each valid which demonstrates the near-tightness (up to a factor of ) of the above bound. For , the ratio can become unbounded. The upper bound is shown constructively and can also be viewed as an approximation algorithm for ATSP with parametrized triangle inequality. We also consider the following problem: in a -triangular graph, does there exist a function such that is bounded above by ? (Here and are the costs of the maximum cost and minimum cost edges respectively.) We show that when , . This upper bound is sharp in the sense that there exist -triangular graphs with . Moreover, for , no such function exists. [Copyright &y& Elsevier]

Abstract: The d-dimensional hypercube, , is the graph on vertices, which correspond to the d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by ) generalizes the notion of hypercubes: The vertices correspond to the d-vectors where the components are from the set , and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the and the . [Copyright &y& Elsevier]

Abstract: A -dimensional box is the Cartesian product where each is a closed interval on the real line. The boxicity of a graph , denoted as , is the minimum integer such that can be represented as the intersection graph of a collection of -dimensional boxes. A unit cube in -dimensional space or a -cube is defined as the Cartesian product where each is a closed interval on the real line of the form . The cubicity of , denoted as , is the minimum integer such that can be represented as the intersection graph of a collection of -cubes. The threshold dimension of a graph is the smallest integer such that can be covered by threshold spanning subgraphs of . In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on vertices with a factor of for any unless . From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on vertices with factor for any unless . In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is -complete to determine whether a given split graph has boxicity at most 3. [ABSTRACT FROM AUTHOR]