Triangle-free Intersection Graphs Research Articles

Coloring triangle-free L-graphs with [formula omitted] colors

It is proved that triangle-free intersection graphs of n L-shapes in the plane have chromatic number O(loglogn). This improves the previous bound of O(logn) (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).

Restricted Frame Graphs and a Conjecture of Scott

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

Coloring Triangle-Free Rectangle Overlap Graphs with $$O(\log \log n)$$ O ( log log n ) Colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ 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 $O(\log\log n)$, improving on the previous bound of $O(\log n)$. 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 $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).

Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

Triangle-free intersection graphs of line segments with large chromatic number

In the 1970s Erdős asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer k we construct a triangle-free family of line segments in the plane with chromatic number greater than k. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.

Coloring [formula omitted]-free intersection graphs of geometric objects in the plane

The intersection graph of a collection C of sets is the graph on the vertex set C, in which C1,C2∈C are joined by an edge if and only if C1∩C2≠0̸. Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (ctlognlogk)clogk, where c is an absolute constant and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk.Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε>0 and for every positive integer t, there exist δ>0 and a positive integer n0 such that every topological graph with n≥n0 vertices, at least n1+ε edges, and no pair of edges intersecting in more than t points, has at least nδ pairwise intersecting edges.

Colouring triangle-free intersection graphs of boxes on the plane

We prove that intersection graphs of boxes on the plane with girth 6 and 8 are 3- and 2-degenerate, respectively. This implies that these graphs are 4- and 3-list-colourable, respectively.