3,160 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 edge-coloring using entropy compression

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    An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio

    Universal targets for homomorphisms of edge-colored graphs

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    A kk-edge-colored graph is a finite, simple graph with edges labeled by numbers 1,,k1,\ldots,k. A function from the vertex set of one kk-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class F\mathcal{F} of graphs, a kk-edge-colored graph H\mathbb{H} (not necessarily with the underlying graph in F\mathcal{F}) is kk-universal for F\mathcal{F} when any kk-edge-colored graph with the underlying graph in F\mathcal{F} admits a homomorphism to H\mathbb{H}. We characterize graph classes that admit kk-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph. For a nonempty graph GG, the density of GG is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of GG. For a nonempty class F\mathcal{F} of graphs, D(F)D(\mathcal{F}) denotes the density of F\mathcal{F}, that is the supremum of densities of graphs in F\mathcal{F}. The main results are the following. The class F\mathcal{F} admits kk-universal graphs for k2k\geq2 if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in F\mathcal{F}. For any such class, there exists a constant cc, such that for any k2k \geq 2, the size of the smallest kk-universal graph is between kD(F)k^{D(\mathcal{F})} and ckD(F)ck^{\lceil D(\mathcal{F})\rceil}. A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (Journal of Algebraic Combinatorics, 8(1):5-13, 1998). One of their results is that for planar graphs, the size of the smallest kk-universal graph is between k3+3k^3+3 and 5k45k^4. Our results yield that there exists a constant cc such that for all kk, this size is bounded from above by ck3ck^3
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