Problems in extremal graph theory

We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of $G$, $\chi(G)$, is the smallest integer $k$ such that $G$ is $k$-colorable. The {\it square} of $G$, written $G^2$, is the supergraph of $G$ in which also vertices within distance 2 of each other in $G$ are adjacent. A graph $H$ is a {\it minor} of $G$ if $H$ can be obtained from a subgraph of $G$ by contracting edges. We show that the upper bound for $\chi(G^2)$ conjectured by Wegner (1977) for planar graphs holds when $G$ is a $K_4$-minor-free graph. We also show that $\chi(G^2)$ is equal to the bound only when $G^2$ contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em $n$-dimensional hypercube}, $Q_n$, is the graph whose vertex set is $\{0,1\}^n$ and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph $G$ that does not contain a forbidden subgraph $H$. We consider the Tur\'an problem where $G$ is $Q_n$ and $H$ is a cycle of length $4k+2$ with $k\geq 3$. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of $Q_n$ over the number of edges of $Q_n$ is $o(1)$, i.e. in the limit this ratio approaches 0 as $n$ approaches infinity

Problems in Extremal Graph Theory

This dissertation consists of six chapters concerning a variety of topics in extremal graph theory.Chapter 1 is dedicated to the results in the papers with Antnio Giro, Gbor Mszros, and Richard Snyder. We say that a graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in eachpair. We study the extremal behavior of maximum degree and diameter in some classes of path-pairable graphs. In particular we show that a path-pairable planar graph must have a vertex of linear degree.In Chapter 2 we present a joint work with Antnio Giro and Teeradej Kittipassorn. Given graphs G and H, we say that a graph F is H-saturated in G if F is H-free subgraph of G, but addition of any edge from E(G) to F creates a copy of H. Here we deal with the case when G is a complete k-partite graph with n vertices in each class, and H is a complete graph on r vertices. We prove bounds, which are tight for infinitely many values of k and r, on the minimal number of edges in a H-saturated graph in G, for this choice of G and H, answering questions of Ferrara, Jacobson, Pfender, and Wenger; and generalizing a result of Roberts.Chapter 3 is about a joint paper with Antnio Giro and Teeradej Kittipassorn. A coloring of the vertices of a digraph D is called majority coloring if no vertex of D receives the same color as more than half of its outneighbours. This was introduced by van der Zypen in 2016. Recently, Kreutzer, Oum, Seymour, van der Zypen, and Wood posed a number of problems related to this notion: here we solve several of them.In Chapter 4 we present a joint work with Antnio Giro. We show that given any integer k there exist functions g1(k), g2(k) such that the following holds. For any graph G with maximum degree one can remove fewer than g1(k) ^{1/2} vertices from G so that the remaining graph H has k vertices of the same degree at least (H) g2(k). It is an approximate version of conjecture of Caro and Yuster; and Caro, Lauri, and Zarb, who conjectured that g2(k) = 0.Chapter 5 concerns results obtained together with Kazuhiro Nomoto, Julian Sahasrabudhe, and Richard Snyder. We study a graph parameter, the graph burning number, which is supposed to measure the speed of the spread of contagion in a graph. We prove tight bounds on the graph burning number of some classes of graphs and make a progress towards a conjecture of Bonato, Janssen, and Roshanbin about the upper bound of graph burning number of connected graphs.In Chapter 6 we present a joint work with Teeradej Kittipassorn. We study the set of possible numbers of triangles a graph on a given number of vertices can have. Among other results, we determine the asymptotic behavior of the smallest positive integer m such that there is no graph on n vertices with exactly m copies of a triangle. We also prove similar results when we replace triangle by any fixed connected graph

Short proofs of some extremal results

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.Comment: 19 page

Breaking Symmetries in Graph Representation

There are many complex combinatorial problems which involve searching for an undirected graph satisfying a certain property. These problems are often highly challenging because of the large number of isomorphic representations of a possible solution. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory

On some interconnections between combinatorial optimization and extremal graph theory

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

Three Existence Problems in Extremal Graph Theory

Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics. In this thesis we take this approach to three different structural questions rooted in extremal graph theory. When studying graph representations, we seek efficient ways to encode the structure of a graph. For example, an {\it interval representation} of a graph $G$ is an assignment of intervals on the real line to the vertices of $G$ such that two vertices are adjacent if and only if their intervals intersect. We consider graphs that have {\it bar $k$-visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations. We obtain results on $\mathcal{F}_k$, the family of graphs having bar $k$-visibility representations. We also study $\bigcup_{k=0}^{\infty} \mathcal{F}_k$. In particular, we determine the largest complete graph having a bar $k$-visibility representation, and we show that there are graphs that do not have bar $k$-visibility representations for any $k$. Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs. Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around $G$ provided that each move transfers weight from a vertex to a heavier neighbor. Our goal in making acquisition moves is to consolidate all of the weight in $G$ on the minimum number of vertices; this minimum number is the {\it acquisition number} of $G$. We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight. We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter $2$. We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex. Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects. Some local conditions are so limiting that very few objects satisfy the requirements. For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor. Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex. We study a related structural restriction where similar phenomena occur. For a fixed graph $H$, we consider those graphs that do not contain $H$ and such that the addition of any edge completes exactly one copy of $H$. Such a graph is called {\it uniquely $H$-saturated}. We study the existence of uniquely $H$-saturated graphs when $H$ is a path or a cycle. In particular, we determine all of the uniquely $C_4$-saturated graphs; there are exactly ten. Interestingly, the uniquely $C_{5}$-saturated graphs are precisely the friendship graphs characterized by Wilf