Disjointness Graphs of Segments

The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large

Disjointness graphs of segments

The disjointness graph G = G (S) of a set of segments S in ℝd 3d ≥ 2, is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies χ(G) ≤ (ω(G))4 + (ω(G)) where ω(G) denotes the clique number of G. It follows, that S has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large. © János Pach, Gábor Tardos, and Géza Tóth

Van Kampen Colimits and Path Uniqueness

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, i.e, categories of objects that can be represented as algebras as well as coalgebras, e.g., the category of directed graphs. Multimodeling requires to construct colimits of models, decomposition is given by pullback. Compositionality requires an exact interplay of these operations, i.e., diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet efficiently checkable condition for the Van Kampen property to hold in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based structural uniqueness feature

Disjointness Graphs of segments in R^2 are almost all Hamiltonian

Let P be a set of n >= 2 points in general position in R^2. The edge disjointness graph D(P) of P is the graph whose vertices are all the closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this note, we give a full characterization of all those edge disjointness graphs that are hamiltonian. More precisely, we shall show that (up to order type isomorphism) there are exactly 8 instances of P for which D(P) is not hamiltonian. Additionally, from one of these 8 instances, we derive a counterexample to a criterion for the existence of hamiltonian cycles due to A. D. Plotnikov in 1998