The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n), and the circumference of a 3-connected claw-free graph is Ω(n). We generalise and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m) edges. We ...

The goal of genome assembly is to reconstruct contiguous portions of a genome (known as contigs) given short reads of DNA sequence obtained in a sequencing experiment. De Bruijn graphs are constructed by finding overlaps of length k − 2 between all substrings of length k − 1 from reads of at least k bases, resulting in a graph where the correct reconstruction of the genome is given by one of th...

A triangulation is said to be even if each vertex has even degree. It is known that every even triangulation on any orientable surface with sufficiently large representativity is 4-colorable [J. Hutchinson, B. Richter, P. Seymour, Colouring Eulerian triangulations, J. Combin. Theory, Ser. B 84 (2002) 225–239], but all graphs on any surface with large representativity are 5-colorable [C. Thomass...

Let R be a commutative ring with identity and )) ( ( R T Γ its total graph. The subject of this article is the investigation of the properties of the corresponding line graph ))). ( ( ( R T L Γ In particular, we determine the girth and clique number of ))). ( ( ( R T L Γ In addition to that, we find the condition for ))) ( ( ( R T L Γ to be Eulerian.

We consider the question whether an infinite eulerian graph has a decomposition into circuits and rays if the graph has only finitely many, say n, vertices of infinite degree, and only finitely many finite components after the removal of the vertices of infinite degree. It is known that the answer is affirmative for n < 2 and negative for n > 4. We settle the remaining case n = 3, showing that ...

We consider an undirected multi(commodity)flow demand problem in which a supply graph is planar, each source-sink pair is located on one of three specified faces of the graph, and the capacities and demands are integer-valued and Eulerian. It is known that such a problem has a solution if the cut and (2,3)-metric conditions hold, and that the solvability implies the existence of an integer solu...

We study the problem of exploring an unknown, strongly connected directed graph. Starting at some node of the graph, we must visit every edge and every node at least once. The goal is to minimize the number of edge traversals. It is known that the competitive ratio of online algorithms for this problem depends on the deficiency d of the graph, which is the minimum number of edges that must be a...

This paper considers the Euler tour of an uncertain random graph, in which some edges exist with some degrees in uncertain measure and others exist with some degrees in probability measure. The concept of Euler index of an uncertain random graph is proposed first, and then a method to calculate the Euler index of an uncertain random graph is given. In addition, some properties of the Euler inde...

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expres...

The simulation of fluid dynamics, typically by numerically solving partial differential equations, is an essential tool in many areas science and engineering. However, the high computational cost can limit application practice may prohibit exploring large parameter spaces. Recent deep-learning approaches have demonstrated potential to yield surrogate models for dynamics. While such exhibit lowe...