Graph Products, Fourier Analysis and Spectral Techniques

Kneser graphs are like Swiss cheese, Discrete Analysis 2018:2, 18 pp. This paper relates two very interesting areas of research in extremal combinatorics: removal lemmas, and of variables. A _removal lemma_ is a result that states that if $S$ is a set that contains only few copies of a certain structure, then one can remove just a few elements from $S$ to create a set $S'$ that has no copies of that structure. For instance, the triangle removal states that a graph with few triangles can be approximated by a graph with no triangles. Removal lemmas play a very important role in extremal and additive combinatorics. As for the of variables, the aspect that is of interest here is that there are many interesting theorems that state that if a function has a certain property, it must be because in a certain sense it depends on only a few variables. For example, a famous result of the first author shows that if a monotone graph property fails to have a sharp threshold, then the property can be approximated by one for which the minimal graphs that satisfy it are all small, with the result that the influence of individual edges on whether the graph has the property is substantial. The main result of this paper is an removal lemma for various families of graphs. It states that if $G$ is a graph in one of these families, then for every set $X$ of vertices in $G$ that spans only a few edges one can throw away just a small proportion of $X$ to obtain a subset $X'$ that is independent. Of course, a result like this is completely false in general: one has only to take a sparse random graph, and then every subset will span few edges but no large subset will be independent. But it is true, for example, for complete bipartite graphs, since a set of vertices that spans a sparse subgraph will have to be contained almost entirely in one of the two vertex sets, and then one can throw away the vertices that are in the other set. One of the families for which the authors prove an edge-removal is that of Kneser graphs. For $0<m<n/2$, the Kneser graph $K(n,m)$ has as its vertex set the set of all subsets of $\{1,2,\dots,n\}$ of size $m$, with two such subsets joined by an edge if and only if they are disjoint. Kneser graphs have large sparse sets, but the natural examples seem to depend on just a few variables: for example, one can take all sets that contain a particular element. (Note that an independent set in $K(n,m)$ is just an intersecting family of $m$-sets.) The authors wanted to prove a precise conjecture that would say that every set of vertices that spans a sparse subgraph can be approximated by a set of vertices that (i) depends on just a few variables and (ii) spans an independent set. Before this paper a weaker statement was known where instead of (ii) one had a sparse set. In this paper the authors settle the conjecture completely by showing that a sparse set of vertices can be approximated by an independent set of vertices. The other family of graphs for which the authors prove an edge removal is that of product graphs. If $H$ is a graph, then the product $G=H^n$ is the graph where $x$ and $y$ are joined if for every coordinate $i$ we have that $x_iy_i$ is an edge of $H$. (There are different definitions of product graphs, but this is the one to which the result applies: note that it makes it quite hard for two vertices to be joined.) For example, if $H$ is a complete graph on $r$ coordinates, then $G=[r]^n$ with two vertices joined if and only if they differ in every coordinate. The proof of the removal uses ideas from Jacob Fox's proof of the triangle removal lemma, which obtains the best known bound for that result. Finally, what has this to do with Swiss cheese? The rough idea is that these graphs have large and highly structured independent sets, and a part of the graph is sparse if and only if it has a big intersection with one of these independent sets. Likewise a Swiss cheese contains lots of large holes, and a region of space contained within the boundary of the cheese will contain only a small amount of cheese if and only if it is almost entirely contained in one of the holes. Article image [by G. J. Wanderer](https://www.flickr.com/photos/106561245@N07/13515457313/in/photolist-mAjfA6-f7mF8i-Tf5Ler-7i4FtY-bi74Nt-gWVXdj-7XoC4N-okRvp5-77HxiZ-7xt4hx-RSUt3B-qeB17D-as2aAc-7K64uq-oL2GMj-6trVsK-UDq452-SP1xSX-d4u57o-fATSUi-8Km2Qs-dJeLzb-qu9Z4Z-dNNvQ2-7eVY4b-6mtCks-frWJEF-aCerNx-7tyjAW-haiMTG-oysrog-9LJVj8-fzno-4VuER1-b5n85M-Jka6K5-a7vHJZ-5S7Yyt-Uwa6kC-dNU7Mm-5AWGod-YSE7a-bzagV7-8tV58z-8YW1XL-5nvECU-QAdC69-8wG84t-TSsHP4-9eoj6S) and distributed under a [CC-BY-NC-SA licence](https://creativecommons.org/licenses/by-nc-sa/2.0/).

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent set. We present properties of different-distance sets, and consider the different-distance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

The crossing number of a graph [Formula: see text], denoted by [Formula: see text], is defined to be the minimum number of crossings that arise among all its drawings in the plane. This concept has been of interest to many researchers who have studied it for many families of graphs. In this paper, we introduce the crossing number of Cartesian product of sunlet graph [Formula: see text] with path [Formula: see text]. Further, we prove that the crossing number of [Formula: see text] is equal to [Formula: see text], along with giving a conjecture for the general case. In addition, we utilize the vertex’s rotation concept in order to prove some necessary conditions for the complete bipartite graph [Formula: see text] to be optimal when it is drawn in the plane, by presenting an upper bound for the crossing number of any subgraph in it together with determining the exact number of crossings in case the vertices of subgraph have the same rotation.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

A finite and simple graph is said to be equitably colorable if its vertices can be partitioned into classes such that each is an independent set and holds for every . The smallest integer for which is equitable chromatic number of and denoted by . The equitable chromatic threshold of a graph , denoted by , is the minimum such that is equitably colorable for all . This paper focuses on the equitable colorability of rooted product of graphs, in particular, exact values or upper bounds of and when and are cycles, paths, complete graphs and complete partite graphs have been found.&#x0D;

The Maximal Independent Set problem is a well-studied problem in the distributed community. We study Maximum and Maximal Independent Set problems on two geometric intersection graphs; interval graphs and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of \(\varOmega (k)\) on the number of rounds whereas \(\varOmega (n)\) is a trivial lower bound on message complexity. Thus our algorithm is both time and message optimal. We also study the maximal independent set problem in bi-interval graphs, a special case of the interval graphs where the intervals have two different lengths. We prove that a maximal independent set can be computed in bi-interval graphs in constant rounds that is \(\frac{1}{6}\)-approximation. For axis-parallel segment intersection graphs, we design an algorithm that finds a maximal independent set in O(D) rounds, where D is the diameter of the graph. We further show that this independent set is a \(\frac{1}{2}\)-approximation. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

The toughness of Kneser graphs

Some new function spaces and their applications to Harmonic analysis

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

Pseudo-one-factorizations of regular graphs of odd order — II. products of graphs

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well‐covered graphs are classified by the Wn property: For a positive integer n, a graph G belongs to class Wn if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of Wn graphs containing arbitrarily large independent sets. A characterization of Wn graphs in terms of well‐covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in Wn graphs.

Maximum weight independent sets and cliques in intersection graphs of filaments

For undirected graphs G=(V,E) and G_0=(V_0,E_0), say that G is a region intersection graph over G_0 if there is a family of connected subsets {R_u \subseteq V_0 : u \in V} of G_0 such that {u,v} \in E \iff R_u \cap R_v \neq \emptyset. We show if G_0 excludes the complete graph K_h as a minor for some h \geq 1, then every region intersection graph G over G_0 with m edges has a balanced separator with at most c_h \sqrt{m} nodes, where c_h is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O(\sqrt{m}). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), and improves over the O(\sqrt{m} \log m) bound of Matousek (2013).