Colouring graphs on surfaces

A hypergraph is a generalization of a graph in which edges—called hyperedges—can connect any number of vertices, not just two. A superhypergraph further extends this framework by introducing recursive powerset constructions, enabling the representation of hierarchical relationships among hyperedges themselves. A topological graph is a graph embedded in the plane, where each vertex is represented by a distinct point and each edge is drawn as a continuous curve connecting its endpoints. A topological hypergraph extends this idea by representing hyperedges as closed curves that enclose sets of vertices on the plane [13]. In this paper, we introduce the concept of a Topological n-SuperHypergraph, which unifies the structural hierarchy of n-superhypergraphs with the geometric intuition of topological hypergraphs. We anticipate that this new formulation will contribute to future developments in graph theory, topology, and network theory.

The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

Graph theoretical studies have been designed to investigate network topologies during life. Network science and graph theory methods may contribute to a better understanding of brain function, both normal and abnormal, throughout developmental stages. The degree to which childhood epilepsies exert a significant effect on brain network organisation and cognition remains unclear. The hypothesis suggests that the formation of abnormal networks associated with epileptogenesis early in life causes a disruption in normal brain network development and cognition, reflecting abnormalities in later life. Neurological diseases with onset during critical stages of brain maturation, including childhood epilepsy, may threaten this orderly neurodevelopmental process. According to the hypothesis that the formation of abnormal networks associated with epileptogenesis in early life causes a disruption in normal brain network development, it is then mandatory to perform a proper examination of children with new-onset epilepsy early in the disease course and a deep study of their brain network organisation over time. In regards, graph theoretical analysis could add more information. In order to facilitate further development of graph theory in childhood, we performed a systematic review to describe its application in functional dynamic connectivity using electroencephalographic (EEG) analysis, focussing on paediatric epilepsy.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

Randomized online graph coloring

Plane graphs and their colorings have been the subject of intensive research since the beginnings of graph theory because of their connection to the fourcolor problem. As stated originally the four-color problem asked whether it is always possible to color the regions of a plane map with four colors such that regions which share a common boundary (and not just a point) receive different colors. The figure on the right shows that coloring the regions of a map is really the same task as coloring the vertices of a plane graph. As in Chapter 12 (page 75) place a vertex in the interior of each region (including the outer region) and connect two such vertices belonging to neighboring regions by an edge through the common boundary.

Graph coloring is an old idea, bringing to mind such celebrated results as the four-color theorem and NP-hardness of the chromatic number of a graph, but not, for most people, the efficient approximation of derivatives. Yet during recent years graph coloring has emerged as a highly effective tool for approximating derivatives, whether Jacobians (matrices of first derivatives of a vector-valued function) or Hessians (matrices of second derivatives of a scalar-valued function). The main issue is how best to exploit the sparsity and symmetry in these matrices to compute them with the least work. Remarkably, the techniques for exploiting sparsity are essentially the same whether the derivatives are approximated by the very old technique of finite differences or computed exactly by the comparatively recent method of automatic differentiation. In both cases, the columns or rows (or both) of a derivative matrix must be partitioned into a small number of groups in order to reduce the number of function evaluations required. It is in formulating, analyzing, and designing algorithms for such problems that graph coloring has proved to be a powerful tool. Indeed, modern software for computing large, sparse Jacobians and Hessians relies on graph coloring algorithms to make the computations feasible. In "What Color Is Your Jacobian?~Graph Coloring for Computing Derivatives," Gebremedhin, Manne, and Pothen give a comprehensive survey of the work in this area during the past 25 years, presenting a unifying framework, several new algorithms, substantial computational results, and a survey of relevant theoretical results.

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Decomposing a planar graph without cycles of length 5 into a matching and a 3-colorable graph

A list of 31 problems presented here reflects some of the main trends in topological graph theory.

An L-list coloring of a graph G is a proper vertex coloring in which every vertex v gets a color from a list L(v) of allowed colors. G is called k-choosable if all lists L(v) have exactly k elements and if G is L-list colorable for all possible assignments of such lists. Verifying conjectures of Erdos, Rubin and Taylor it was shown during the last years that every planar graph is 5-choosable and that there are planar graphs which are not 4-choosable. The question whether there are 3-colorable planar graphs which are not 4-choosable remained unsolved. The smallest known example far a non-4-choosable planar graph has 75 vertices and is described by Gutner. In fact, this graph is also 3 colorable and answers the above question. In addition, we give a list assignment for this graph using 5 colors only in all of the lists together such that the graph is not List-colorable. © 1997 John Wiley & Sons, Inc.

A generalization of a theorem of de Bruijn and Erdös on the chromatic number of infinite graphs

Given an arbitrary graph, wh~t can we say about the setl of all chromatic numbers of its subgraphs ? In particular, for e~rdinals m ~ n, does every graph of chromatic number m have a subgraph of chromatic number n? For n ~ ~t 0 the affirmative answer follows from the theorem of DE BI~UIJ~ and ERD6S [1], SO the simplest nontrivial question is whether every graph of chromatic number ~2 has a subgraph of chromatic number Nr I have no positive results; the purpose of this note is only to state the probleml in the form of 2 dubious conjectures, and to prove that the stronger conjecture implies a weak form of the GCH. If % is a set, [S] 2 = {X c~ S " [X l = 2}. We regard a graph as a system G = (V, E) consisting of a set V of vertices and a set E ~c [ V] ~ of edges. A coloring of G is a function f defined on V such thatf(x) :~f(y) whenever {x, y} # E. G is m-colorable if there is a coloring f such that If(V) l ~ m. The chromatic number of G, ehr(G), is the least m such that G is m-eolorable. A graph G" ---- (V', E') is a vertex subgraph of G if V' ~ V and E' = E N [V'~]; an edge subgraph if V' ~ V and E' c E. ConJEcturE 1. For m ~ n, every graph of chromatic number m has a vertex subgraph of chromatic number n.

Four-color theorem has only been proved by computer since it was proposed, many people have proposed their mathematical proof of four-color theorem, but their proof is disputed then, what lead to a situation that the mathematical proof of four-color theorem has been lacking to this day. In this article, we have summarized some laws based on previous researches, and proposed a mathematical proof of four-color theorem by using these laws trough a recursive method.