A Note on the Seven Bridges of Königsberg Problem

Locally self-avoiding Eulerian tours

Recently, graph contrastive learning (GCL) has achieved remarkable performance in graph representation learning. However, existing GCL methods usually follow a dual-channel encoder network (i.e., Siamese networks), which adds to the complexity of the network architecture. Additionally, these methods overly depend on varied data augmentation techniques, corrupting graph information. Furthermore, they are heavily reliant on large quantities of negative nodes for each object node, which requires tremendous memory costs. To address these issues, we propose a novel and simple graph representation learning framework, named SimGRL. Firstly, our proposed network architecture only contains one encoder based on a graph neural network instead of a dual-channel encoder, which simplifies the network architecture. Then we introduce a distributor to generate triplets to obtain the contrastive views between nodes and their neighbors, avoiding the need for data augmentations. Finally, we design a triplet loss based on adjacency information in graphs that utilizes only one negative node for each object node, reducing memory overhead significantly. Extensive experiments demonstrate that SimGRL achieves competitive performance on node classification and graph classification tasks, especially in terms of running time and memory overhead.

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

Graph theory is a branch of mathematics that defines the study of graphs and is now a significant asset to analyzing various systems in various fields. This chapter gives an overview of the subject which includes vertices, edges, paths, cycles, and others like trees, bipartite, complete and planar graphs. Common measures and descriptors, including vertex degrees, graph connection, components, and adjacency and incidence maps, are considered to form a reference base for graph structures. The chapter also covers special concepts like Eulerian paths & ways and its focus on these types of recovery are blow up by their importance in routing, scheduling & network analysis. Also, current developments such as random graphs, small-world networks and scale-free networks are presented to illustrate the applicability of graph theory in computing gigantic and connected frameworks. This chapter, through its examples of mathematical theories addressing problems in various fields of science and technology, explains and reiterates the flexibility of graph theory in solving real-world issues.

Known as a branch of Discrete Mathematics (DM), Graph Theory (GT) describes and solves problems of discrete nature through nodes (i.e., vertices) and arcs (i.e., edges). In this regard, a prominent problem is to find the Eulerian circuits. This paper indicates that the problem can be analyzed through operations research methods. In more general terms, finding the Eulerian circuits could be considered a pathfinding problem. Hence, this paper proposes a pure binary mathematical model to describe the relationship between the variables employed to find the Eulerian circuits. All the analyses in this paper were performed in MATLAB. The proposed model can be solved by many optimization software applications. Finally, several numerical examples are presented and solved through the proposed method. All the analyses in this paper were performed in MATLAB. This paper indicated that the problem(Eulerian Circuits in Complete Graphs) could be studied and solved from the perspective of operations research.

Alternating Knot Diagrams, Euler Circuits and the Interlace Polynomial

A characterisation of graphs having three pairwise compatible Euler tours

The Graceful Tree Conjecture stated by Rosa in the mid 1960s says that every tree can be gracefully labelled. It is one of the best known open problems in Graph Theory. The conjecture has caused a great interest in the study of gracefulness of simple graphs and has led to many new contributions to the list of graceful graphs. However, it has to be acknowledged that not much is known about the structure of graceful graphs after 55 years. Our paper adds an infinite family of classes of graceful graphs to the list of known simple graceful graphs. We introduce classes of \(k\)-enriched fan graphs \(kF_n\) for all integers \(k, n\ge 2\) and we prove that these graphs are graceful. Moreover, we provide characterizations of the \(k\)-enriched fan graphs \(kF_n\) among all simple graphs via Sheppard's labelling sequences introduced in the 1970s, as well as via labelling relations and graph chessboards. These last approaches are new tools for the study of graceful graphs introduced by Haviar and Ivaska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a nice visualization of graceful labellings. We close our paper with an open problem concerning another infinite family of extended fan graphs.

A graph G = (V, E) is said to be weakly four‐connected if G is 4‐edge‐connected and G – x is 2‐edge‐connected for every x ∈ V. We prove that every weakly four‐connected Eulerian graph has a 2‐connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank . © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 230–242, 2006

Se presentan tres algoritmos para la búsqueda de caminos orientados en un digrafo, basados en la generación de un árbol en el que se hace una búsqueda exhaustiva, en amplitud en el primer algoritmo, y en profundidad en el segundo y en el tercero. El primero permite encontrar todos los caminos óptimos entre dos vértices; el segundo permite resolver este mismo problema así como el de hallar los caminos hamiltonianos con origen en un vértice, o los ciclos de cualquier orden, en tanto que el tercero permite encontrar todos los caminos o circuitos eulerianos. Se describen, asimismo, dos algoritmos que hacen uso del mismo tipo de técnicas para el análisis de la conectividad de un grafo. El primero permite separar un grafo no conexo en sus partes conexas, y el segundo permite la detección de puentes en grafos conexos.

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.

Graph representation learning has made major strides over the past decade. However, in many relational domains, the input data are not suited for simple graph representations as the relationships between entities go beyond pairwise interactions. In such cases, the relationships in the data are better represented as hyperedges (set of entities) of a non-uniform hypergraph. While there have been works on principled methods for learning representations of nodes of a hypergraph, these approaches are limited in their applicability to tasks on non-uniform hypergraphs (hyperedges with different cardinalities). In this work, we exploit the incidence structure to develop a hypergraph neural network to learn provably expressive representations of variable sized hyperedges which preserve local-isomorphism in the line graph of the hypergraph, while also being invariant to permutations of its constituent vertices. Specifically, for a given vertex set, we propose frameworks for (1) hyperedge classification and (2) variable sized expansion of partially observed hyperedges which captures the higher order interactions among vertices and hyperedges. We evaluate performance on multiple real-world hypergraph datasets and demonstrate consistent, significant improvement in accuracy, over state-of-the-art models.

Self-supervised multiplex graph representation learning (SMGRL) aims to capture the information from the multiplex graph, and generates discriminative embedding without labels. However, previous SMGRL methods still suffer from the issues of efficiency and effectiveness due to the processes, e.g., data augmentation, negative sample encoding, complex pretext tasks, etc. In this paper, we propose a simple method to achieve efficient and effective SMGRL. Specifically, the proposed method removes the processes (i.e., data augmentation and negative sample encoding) for the SMGRL and designs a simple pretext task, for achieving the efficiency. Moreover, the proposed method also designs an intra-graph decorrelation loss and an inter-graph decorrelation loss, respectively, to capture the common information within individual graphs and the common information across graphs, for achieving the effectiveness. Extensive experimental results verify the efficiency and effectiveness of our method, compared to 11 comparison methods on 4 public benchmark datasets, on the node classification task.

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today. Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs.