12 research outputs found

    Improved bounds on coloring of graphs

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    Given a graph GG with maximum degree Δ3\Delta\ge 3, we prove that the acyclic edge chromatic number a(G)a'(G) of GG is such that a(G)9.62(Δ1)a'(G)\le\lceil 9.62 (\Delta-1)\rceil. Moreover we prove that: a(G)6.42(Δ1)a'(G)\le \lceil 6.42(\Delta-1)\rceil if GG has girth g5g\ge 5\,; a'(G)\le \lceil5.77 (\Delta-1)\rc if GG has girth g7g\ge 7; a'(G)\le \lc4.52(\D-1)\rc if g53g\ge 53; a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil. We further prove that the acyclic (vertex) chromatic number a(G)a(G) of GG is such that a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the star-chromatic number χs(G)\chi_s(G) of GG is such that \chi_s(G)\le \lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic number \chi^\b(G) of GG is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are decreasing functions of \b such that k_1(\b)\in[4, 6] and k_2(\b)\in[2,5]. To obtain these results we use an improved version of the Lov\'asz Local Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of Theorem 2 (items c-f) written in more detail

    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 list edge coloring of outerplanar graphs

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    AbstractAn acyclic list edge coloring of a graph G is a proper list edge coloring such that no bichromatic cycles are produced. In this paper, we prove that an outerplanar graph G with maximum degree Δ≥5 has the acyclic list edge chromatic number equal to Δ

    Acyclic Edge Coloring of Graphs with Maximum Degree 4

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    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 is denoted by a'(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a'(G) <= Delta+2, where Delta=Delta(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Delta(G)<= 4, with the additional restriction that m <= 2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m <= 2n, when Delta(G)<= 4. It follows that for any graph G if Delta(G)<= 4, then a'(G) <= 7
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