1,273 research outputs found

    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 Subgraphs of Planar Digraphs

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    An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on nn vertices without directed 2-cycles possesses an acyclic set of size at least 3n/53n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if gg is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least (1−3/g)n(1 - 3/g)n.Comment: 9 page

    Complete Acyclic Colorings

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    We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
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