Abstract
Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G)=k, then χ(G)≤(k+12). If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G)≤k+12(k+23). Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor.
Highlights
Given a family of sets, C, the intersection graph of C is the graph, whose vertices correspond to the elements of C, and two vertices are joined by an edge if the corresponding sets have a nonempty intersection
The disjointness graph of C is the complement of the intersection graph of C, that is, two vertices are joined by an edge if the corresponding sets are disjoint
There are many interesting results connecting the clique number and the chromatic number of geometric intersection graphs, starting with a beautiful theorem of Asplund and Grünbaum [2], which states that every intersection graph G of axis-parallel rectangles in the plane satisfies χ(G) ≤ 4(ω(G)
Summary
Given a family of sets, C, the intersection graph of C is the graph, whose vertices correspond to the elements of C, and two vertices are joined by an edge if the corresponding sets have a nonempty intersection. Computing the chromatic number of the disjointness graph of a family of objects, C, is equivalent to determining the clique cover number of the corresponding intersection graph G, that is, the minimum number of cliques whose vertices together cover the vertex set of G. This problem can be solved in polynomial time only for some very special families (for instance, if C consists of intervals along a line or arcs along a circle [13]). There is a vast literature providing approximation algorithms or inapproximability results for the clique cover number [8, 9]
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