713 research outputs found

    Strong edge-colorings for k-degenerate graphs

    Full text link
    We prove that the strong chromatic index for each kk-degenerate graph with maximum degree Δ\Delta is at most (4k2)Δk(2k1)+1(4k-2)\Delta-k(2k-1)+1

    K3K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

    Get PDF
    A K3K_3-WORM coloring of a graph GG is an assignment of colors to the vertices in such a way that the vertices of each K3K_3-subgraph of GG get precisely two colors. We study graphs GG which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a K3K_3-WORM coloring with two colors. In fact for every integer k3k\ge 3 there exists a K3K_3-WORM colorable graph in which the minimum number of colors is exactly kk. There also exist K3K_3-WORM colorable graphs which have a K3K_3-WORM coloring with two colors and also with kk colors but no coloring with any of 3,,k13,\dots,k-1 colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide kk-colorability for every k2k \ge 2 (and remains intractable even for graphs of maximum degree 9 if k=3k=3). On the other hand, we prove positive results for dd-degenerate graphs with small dd, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.Comment: 18 page

    Inside-Out Polytopes

    Get PDF
    We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat

    Strong chromatic index of sparse graphs

    Full text link
    A coloring of the edges of a graph GG is strong if each color class is an induced matching of GG. The strong chromatic index of GG, denoted by χs(G)\chi_{s}^{\prime}(G), is the least number of colors in a strong edge coloring of GG. In this note we prove that χs(G)(4k1)Δ(G)k(2k+1)+1\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1 for every kk-degenerate graph GG. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that % \chi_{s}^{\prime}(G)\leq 4\Delta -3 for any chordless graph GG. Both bounds remain valid for the list version of the strong edge coloring of these graphs

    Linear Transformations Between Colorings in Chordal Graphs

    Get PDF
    Let k and d be such that k >= d+2. Consider two k-colorings of a d-degenerate graph G. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. If k=d+2, we know that there exists graphs for which a quadratic number of recolorings is needed. And when k=2d+2, there always exists a linear transformation. In this paper, we prove that, as long as k >= d+4, there exists a transformation of length at most f(Delta) * n between any pair of k-colorings of chordal graphs (where Delta denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two k-colorings c_1,c_2 computes a linear transformation between c_1 and c_2

    Conflict-Free Coloring Made Stronger

    Full text link
    In FOCS 2002, Even et al. showed that any set of nn discs in the plane can be Conflict-Free colored with a total of at most O(logn)O(\log n) colors. That is, it can be colored with O(logn)O(\log n) colors such that for any (covered) point pp there is some disc whose color is distinct from all other colors of discs containing pp. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of nn discs in the plane can be colored with a total of at most O(klogn)O(k \log n) colors such that (a) for any point pp that is covered by at least kk discs, there are at least kk distinct discs each of which is colored by a color distinct from all other discs containing pp and (b) for any point pp covered by at most kk discs, all discs covering pp are colored distinctively. We call such a coloring a {\em kk-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of nn simple closed Jordan regions with union-complexity bounded by O(n1+α)O(n^{1+\alpha}), we prove that there exists a kk-Strong Conflict-Free coloring with at most O(knα)O(k n^\alpha) colors. \item [(iii)] We prove that any set of nn axis-parallel rectangles can be kk-Strong Conflict-Free colored with at most O(klog2n)O(k \log^2 n) colors. \item [(iv)] We provide a general framework for kk-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of kk-Strong Conflict-Free coloring and the recently studied notion of kk-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings
    corecore