Circulant tournaments of prime order are tight

AbstractWe say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p≥3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory

Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.Comment: 33 pages, 8 Figure

DIGRAPH GROUPS AND RELATED GROUPS

This thesis investigates finite digraph groups and related groups like the generalization of Johnson and Mennicke groups. Cuno and Williams introduced the term "digraph group" for the first time in [9], 2020. The groups are defined by non-empty presentations and each relator is in the form R(x, y), where x and y are distinct generators and R(.,.) is defined by some fixed cyclically reduced word R(a, b) that involves both a and b. There is a directed graph associated with each of these presentations, where the vertices correspond to the generators and the arcs correspond to the relators. In Chapter 2, we investigate Cayley digraph groups to determine whether they are finite cyclic and provide formulae to calculate the order. In Chapters 3 and 4, the girth of the underlying undirected graph is at least 4. We show that the resulting groups are non-trivial and cannot be finite of rank 3 or higher under the condition |V|=|A|-1 in Chapter 3. We investigate when the corresponding digraph groups are finite cyclic for |V| \leq |A| in Chapter 4 and we are able to show that the corresponding group of strongly connected and semi-connected digraphs under certain standard conditions which are known to be necessary for the digraph group to be finite ((i)-(iv) defined in Preamble 4.1). We generalise Johnson and Mennicke groups, which are non-cyclic finite groups defined in terms of a digraph that is a directed triangle to digraphs that are n-vertex tournaments in Chapter 5. In Chapter 6 we use GAP to perform a computational investigation into digraph groups with particular relators and we obtain results whether the corresponding digraph groups are cyclic, abelian, perfect or not. We also provide their size, derived series, derived length and facts about isomorphism between them. The relators used correspond to the those used in the Mennicke and Johnson groups, and some new fixed relators. We obtain digraph presentations of various 2-groups, 3-groups and perfect groups

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

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

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200