1,588 research outputs found

    Transitive and Gallai colorings

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    A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs, and generalize both notions to Coxeter systems, matroids and commutative algebras. It is shown that for any finite matroid (or oriented matroid), the maximal number of colors is equal to the matroid rank. This generalizes a result of Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or transitive) colorings of the matroid that use at most kk colors is a polynomial in kk. Also, for any acyclic oriented matroid, represented over the real numbers, the number of transitive colorings using at most 2 colors is equal to the number of chambers in the dual hyperplane arrangement. We count Gallai and transitive colorings of the root system of type A using the maximal number of colors, and show that, when equipped with a natural descent set map, the resulting quasisymmetric function is symmetric and Schur-positive.Comment: 31 pages, 5 figure

    On the phase transitions of graph coloring and independent sets

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    We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number

    A bivariate chromatic polynomial for signed graphs

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    We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial cΓ(k,l)c_\Gamma(k,l) which counts all (k+l)(k+l)-colorings of a graph Γ\Gamma such that adjacent vertices get different colors if they are ≤k\le k. Our first contribution is an extension of cΓ(k,l)c_\Gamma(k,l) to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for cΓ(k,l)c_\Gamma(k,l) and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure

    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 k≥2k\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 k≥2k \geq 2, the size of the smallest kk-universal graph is between kD(F)k^{D(\mathcal{F})} and ck⌈D(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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