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

On the Chromatic Number of Disjointness Graphs of Curves

Let omega(G) and chi(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 omega(G)=k, then chi(G)<= binom{k+1}{2}. If we only require that every curve is x-monotone and intersects the y-axis, then we have chi(G)<= k+1/2 binom{k+2}{3}. 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 K_k-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Omega(k^{4}) colors. This matches the upper bound up to a constant factor

The Maximum Chromatic Number of the Disjointness Graph of Segments on nn-point Sets in the Plane with n≤16n\leq 16

Let $P$ be a finite set of points in general position in the plane. The disjointness graph of segments $D(P)$ 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. As usual, we use $\chi(D(P))$ to denote the chromatic number of $D(P)$, and use $d(n)$ to denote the maximum $\chi(D(P))$ taken over all sets $P$ of $n$ points in general position in the plane. In this paper we show that $d(n)=n-2$ if and only if $n\in \{3,4,\ldots ,16\}$.Comment: 25 pages, 3 figure

Colouring Polygon Visibility Graphs and Their Generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time

Coloring polygon visibility graphs and their generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo- polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 · 4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo- visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a coloring with the claimed number of colors can be computed in polynomial time