234,027 research outputs found
Coloring triangle-free rectangle overlap graphs with colors
Recently, it was proved that triangle-free intersection graphs of line
segments in the plane can have chromatic number as large as . Essentially the same construction produces -chromatic
triangle-free intersection graphs of a variety of other geometric
shapes---those belonging to any class of compact arc-connected sets in
closed under horizontal scaling, vertical scaling, and
translation, except for axis-parallel rectangles. We show that this
construction is asymptotically optimal for intersection graphs of boundaries of
axis-parallel rectangles, which can be alternatively described as overlap
graphs of axis-parallel rectangles. That is, we prove that triangle-free
rectangle overlap graphs have chromatic number , improving on
the previous bound of . To this end, we exploit a relationship
between off-line coloring of rectangle overlap graphs and on-line coloring of
interval overlap graphs. Our coloring method decomposes the graph into a
bounded number of subgraphs with a tree-like structure that "encodes"
strategies of the adversary in the on-line coloring problem. Then, these
subgraphs are colored with colors using a combination of
techniques from on-line algorithms (first-fit) and data structure design
(heavy-light decomposition).Comment: Minor revisio
Dominator Chromatic Number of Circular-Arc Overlap Graphs
A dominator coloring problem of the graph G is a proper coloring of the graph, where every vertex of the graph inevitably dominates an entire color class. It is observed that the number of color classes selected is of minimum order and when only it suits to be the dominator chromatic number. The present paper brings to the fore, the findings related to dominator chromatic number of some special classes of circular-arc overlap graphs, upholding the concepts on the bounds of dominator chromatic number and the relation between the chromatic number and the dominator chromatic number. Keywords: Circular-arc overlap graphs, independent set, chromatic number, dominator chromatic numbe
On the chromatic number of multiple interval graphs and overlap graphs
AbstractLet χ(G) and ω(G) denote the chromatic number and clique number of a graph G. We prove that χ can be bounded by a function of ω for two well-known relatives of interval graphs. Multiple interval graphs (the intersection graphs of sets which can be written as the union of t closed intervals of a line) satisfy χ⩽2t(ω−1) for ω⩾2. Overlap graphs satisfy χ⩽2ωω2(ω−1)
Asymptotic behavior of the number of Eulerian orientations of graphs
We consider the class of simple graphs with large algebraic connectivity (the
second-smallest eigenvalue of the Laplacian matrix). For this class of graphs
we determine the asymptotic behavior of the number of Eulerian orientations. In
addition, we establish some new properties of the Laplacian matrix, as well as
an estimate of a conditionality of matrices with the asymptotic diagonal
predominanceComment: arXiv admin note: text overlap with arXiv:1104.304
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