Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

On embeddings of CAT(0) cube complexes into products of trees

We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $\Delta$ can be coloured with at most $\epsilon(\Delta)=M\Delta^{26}$ colours, for a fixed constant $M$. This implies that ${\bf X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $\epsilon(\Delta)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $\epsilon(\Delta)$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the computation of the bounds in Theorem 1. Some figures repaire

Polychromatic Colorings on the Hypercube

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.Comment: 24 page

Isometric Embeddings in Trees and Their Use in Distance Problems

International audienceWe present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows: (1) We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5−ε}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs. (2) Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an O(n^{5/3})-time algorithm for computing all the eccentricities). (3) We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2. (4) Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, Bergé and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem. Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate

Distance Labeling Schemes for Cube-Free Median Graphs

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. One of the important problems is finding natural classes of graphs admitting distance labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance labeling scheme with labels of O(log^3 n) bits

Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs

On sparse graphs, Roditty and Williams [2013] proved that no O(n^{2-?})-time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n^{1.6456}log^{O(1)} n)