If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980’s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.

A survey of χ‐boundedness

We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "on-cas" behaviour -- obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity.Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973075.19

Testing planarity of partially embedded graphs

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

In the 1970s, Erdos 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.

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertex separator of size O(root m log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K-t not subset of G for some t, then the chromatic number of G is at most (log n)(O(log t)); (ii) if K-t,K-t not subset of G, then G has at most t(log t)(O(1)n) edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.

/pdf/applications-of-a-new-separator-theorem-for-string-graphs-59daw04wwd.pdf

Applications of a New Separator Theorem for String Graphs

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear.

We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbic's comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0,1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0,1] becomes NP-complete.

Extending partial representations of function graphs and permutation graphs

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents, point by point, a partially ordered set. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. There are many recent results about variants of the general on-line chain partition problem. With this survey we attempt to give an overview over the state of the art in this field. As particularly interesting aspects of the article we see:

https://www.dfki.de/web/forschung/publikationen/renameFileForDownload?filename=survey.pdf&file_id=uploads_1837

On-Line Chain Partitions of Orders: A Survey

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $$X$$X in $$\mathbb{R }^2$$R2 that is not an axis-aligned rectangle and for any positive integer $$k$$k produces a family $$\mathcal{F }$$F of sets, each obtained by an independent horizontal and vertical scaling and translation of $$X$$X, such that no three sets in $$\mathcal{F }$$F pairwise intersect and $$\chi (\mathcal{F })>k$$ź(F)>k. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

/pdf/triangle-free-geometric-intersection-graphs-with-large-3gtuzahllr.pdf

Tomasz Krawczyk

Papers

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

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

Extending partial representations of function graphs and permutation graphs

On-Line Chain Partitions of Orders: A Survey

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number