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    Acyclic edge coloring of graphs

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    An {\em acyclic edge coloring} of a graph GG is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} \chiup_{a}'(G) of a graph GG is the least number of colors needed in an acyclic edge coloring of GG. Fiam\v{c}\'{i}k (1978) conjectured that \chiup_{a}'(G) \leq \Delta(G) + 2, where Δ(G)\Delta(G) is the maximum degree of GG. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph GG with maximum degree at most κ\kappa is {\em κ\kappa-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H) \leq \kappa for every proper subgraph HH of GG. The purpose of this paper is to provide many structural lemmas on κ\kappa-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (\autoref{NMAD4}). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every 55-cycle has at most three edges contained in triangles (\autoref{NoAdjacent}), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph GG without intersecting triangles satisfies \chiup_{a}'(G) \leq \Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case and prove it: if GG is a graph with Δ(G)≥3\Delta(G) \geq 3 and all the 3+3^{+}-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope the structural lemmas will shed some light on the acyclic edge coloring problems.Comment: 19 page

    Acyclic edge coloring of subcubic graphs

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    AbstractAn 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
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