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تسجيل مستخدم جديدColoring translates and homothets of a convex body
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We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in $RR^n$.
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For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes $G^{text{hom}}(A)$ and $G^{text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, resp
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We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rado-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex plan
er domain; and Tuttes embedding theorem for planar graphs - RKCs discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.
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