Related papers: An introduction to graph theory

Graphs are extremely versatile and ubiquitous mathematical structures with potential to model a wide range of domains. For this reason, graph problems have been of interest since the early days of computer science. Some of these problems…

Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the…

This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial…

In this paper we study fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. To prepare the ground, we define various types of subhypergraphs, as…

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$…

We survey some uses of adjoint functors in graph theory pertaining to colourings, complexity reductions, multiplicativity, circular colourings and tree duality. The exposition of these applications through adjoint functors unifies the…

Graphs have a superior ability to represent relational data, like chemical compounds, proteins, and social networks. Hence, graph-level learning, which takes a set of graphs as input, has been applied to many tasks including comparison,…

We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and…

We extend the closed graph theorem and the open mapping theorem to a context in which a natural duality interchanges their extensions.

Graph machine learning has been extensively studied in both academia and industry. However, in the literature, most existing graph machine learning models are designed to conduct training with data samples in a random order, which may…

This chapter presents an introduction to graph colouring algorithms. The focus is on vertex-colouring algorithms that work for general classes of graphs with worst-case performance guarantees in a sequential model of computation. The…

A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary…

We present a framework for representing and modeling data on graphs. Based on this framework, we study three typical classes of graph signals: smooth graph signals, piecewise-constant graph signals, and piecewise-smooth graph signals. For…

Entanglement is a complexity measure of digraphs that origins in fixed-point logics. Its combinatorial purpose is to measure the nested depth of cycles in digraphs. We address the problem of characterizing the structure of graphs of…

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have…

Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…

We present a first algorithm for finding Euler tours in undirected graphs in the StrSort model. This model is a relaxation of the semi streaming model. The graph is given as a stream of its edges and can only be read sequentially, but while…

Various connections between the theory of permutation groups and the theory of topological groups are described. These connections are applied in permutation group theory and in the structure theory of topological groups. The first draft of…

Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…