# Manual Eulerian Graphs and Related Topics: Part 1, Volume 1

Now let's count the same angles the other way. Each interior vertex is surrounded by triangles and contributes a total angle of 2 pi to the sum. The vertices on the outside face contribute 2 pi - theta v. The total exterior angle of any polygon is 2 pi, so the total angle is 2 pi V - 4 pi. This is the method used by Descartes in Sommerville attributes this proof to Lhuilier and Steiner. Hilton and Pederson use angles in a similar way to relate the Euler characteristic of a polyhedral surface to its total angular defect.

Proof 9: Spherical Angles The proof by sums of angles works more cleanly in terms of spherical triangulations, largely because in this formulation there is no distinguished "outside face" to cause complications in the proof. Now perform a similar light-shining experiment to the one described on the index page : place a light source at an interior point of the polyhedron, and place a spherical screen outside the polyhedron having the light source as its centerpoint.

The shadows cast on the screen by the polyhedron edges will form a spherical triangulation. Adding the same angles another way, in terms of the vertices, gives a total of 2 V pi. Sommerville attributes this proof to Legendre.

## Eulerian path

Because of its connections with geometric topology , this is the proof used by Weeks , who also gives an elegant proof of the spherical angle-area relationship based on inclusion-exclusion of great-circular double wedges. Proof Pick's Theorem We have translated our sum-of-angles proof to spherical trigonometry , in the process obtaining formulas in terms of sums of areas of faces.

Now we examine similar formulas for sums of areas in planar geometry, following a suggestion of Wells. This can be proven in a number of ways, for instance by choosing a horizontal line L passing below the polygon and partitioning the polygon's area into the sum of signed areas of trapezoids from L to each edge. These proofs do not require Euler's formula so there is no danger of circular reasoning.

First draw the planar graph corresponding to the polygon, with straight line segment edges. Then move every vertex to the nearest integer vertex; the result is an equivalent planar graph drawn in the grid. The outer face of the graph is covered exactly once by the remaining faces, so the sum of the areas of the remaining faces should equal the area of the outer faces. A vertex interior to the graph contributes a term to S equal to its degree, whereas a vertex on the outer face contributes only its degree minus one. Unfortunately Pick's theorem does not generalize to higher dimensions, so this approach seems unlikely to work for proving higher-dimensional forms of Euler's formula.

Proof Ear Decomposition A graph is two-edge-connected if removing any edge leaves a connected subgraph. Two-edge-connectivity is equivalent to the existence of an ear decomposition : a partition of the edges of the graph into a sequence of ears simple paths and cycles , with the first ear being a single vertex; the start and end of each successive ear should be vertices occurring in previous ears, but all other vertices in an ear should be new.

Such a decomposition can be found one ear at a time: start each ear by any unused edge e from an already-explored vertex, and continue by a shortest path back to another already-explored vertex such a path must exist because e cannot disconnect the graph. We can use such a decomposition in a proof of the Euler formula for polyhedra: The graph G of a polyhedron is two-edge-connected, since if we remove an edge e we can still connect its endpoints by a path around the other side of one of the two faces of G containing e.

Find an ear decomposition of G, and consider the process of forming G by adding ears one at a time starting from a single vertex. Initially there is one vertex, one face, and no edges. Each new ear forms a path connecting two vertices on the boundary of a face, splitting the face in two; the path has one more edge than it has vertices.

So if the ear has k edges, its addition increases V by k -1, E by k , and F by 1, and the result follows by induction on the number of ears. Ear decompositions have been especially useful in the design of parallel algorithms, since they are easier to find in parallel than are other structural decompositions of graphs such as depth first search trees. For an example of this see my work on recognizing series parallel graphs.

Proof Shelling Ziegler interprets a polyhedron or polytope as a complex of polyhedral faces of varying dimensions. Ziegler defines a shelling of a complex of polyhedral faces to be a linear ordering on the maximal-dimension faces facets such that, if the facets are of dimension at least one, the ordering satisfies the following properties: The boundary of the first facet F 0 has a shelling.

Every convex polytope has a shelling found by traveling in a straight line from some point near one face of the polyhedron, and ordering the faces by the sequence of points at which the line crosses the plane of a face and the face becomes visible. The line must be imagined to pass "to infinity and beyond" through to the other side of the polyhedron. The intersection of a facet F j with previous facets can be found geometrically as the portion of the boundary of the facet visible from the intersection point of the viewing line with the plane of the facet; this shows that the lower-dimensional shelling required by property 2 is of the same geometric type.

Now if P is a d -polytope with shelling order F 1 ,F 2 , Removing the two "extra" faces f -1 and f 3 from this sum gives us the usual Euler formula. Proof Triangle Removal This proof is really just a variation on shelling , but is included here for its historical significance: it was used by Cauchy, and was examined at length by Lakatos.

Begin with a convex planar drawing of the polyhedron's edges. If there is a non-triangular face, add a diagonal to a face, dividing it in two and adding one to the numbers of edges and faces; the result then follows by induction. So suppose we have any planar graph with all interior faces triangular, with at least two such faces, and with the further property that one can get from any interior point to any other by a path that avoids the boundary of the graph's outer face.

The triangulation of the convex drawing of our polyhedron clearly satisfies these properties. Then there are always at least two triangles having edges on this boundary, such that the removal of either one leaves a single triangle or a smaller graph of the same type; this can be proved by an induction on the number of triangles, for if some boundary triangle disconnects the interior points, the two disconnected components on its two non-boundary edges must either be single triangles which are removable or have by induction two removable boundary triangles, at least one of which will be removable in the overall graph.

So remove boundary triangles one by one; at each step we remove either one edge and one face, or two edges, a face, and a vertex.

Define a height function on the surface of the polyhedron as follows: Choose arbitrary heights for each vertex. Hierholzer 's paper provides a different method for finding Euler cycles that is more efficient than Fleury's algorithm:. The formula states that the number of Eulerian circuits in a digraph is the product of certain degree factorials and the number of rooted arborescences. The latter can be computed as a determinant , by the matrix tree theorem , giving a polynomial time algorithm.

BEST theorem is first stated in this form in a "note added in proof" to the Aardenne-Ehrenfest and de Bruijn paper The original proof was bijective and generalized the de Bruijn sequences. It is a variation on an earlier result by Smith and Tutte Counting the number of Eulerian circuits on undirected graphs is much more difficult. This problem is known to be P-complete. The asymptotic formula for the number of Eulerian circuits in the complete graphs was determined by McKay and Robinson : . A similar formula was later obtained by M.

Isaev for complete bipartite graphs : . Eulerian trails are used in bioinformatics to reconstruct the DNA sequence from its fragments. In an infinite graph , the corresponding concept to an Eulerian trail or Eulerian cycle is an Eulerian line, a doubly-infinite trail that covers all of the edges of the graph. It is not sufficient for the existence of such a trail that the graph be connected and that all vertex degrees be even; for instance, the infinite Cayley graph shown, with all vertex degrees equal to four, has no Eulerian line. For an infinite graph or multigraph G to have an Eulerian line, it is necessary and sufficient that all of the following conditions be met:  .

This multigraph is not Eulerian, therefore, a solution does not exist. Every vertex of this graph has an even degree. Therefore, this is an Eulerian graph. In graph theory, an Eulerian trail or Eulerian path is a trail in a finite graph that visits every edge exactly once allowing for revisiting vertices. The problem can be stated mathematically like this: Given the graph in the image, is it possible to construct a path or a cycle, i.

Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without pr. One possible Hamiltonian cycle through every vertex of a dodecahedron is shown in red — like all platonic solids, the dodecahedron is Hamiltonian The above as a two-dimensional planar graph The Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle.

A possible Hamiltonian path is shown. In the mathematical field of graph theory, a Hamiltonian path or traceable path is a path in an undirected or directed graph that visits each vertex exactly once.

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A Hamiltonian cycle or Hamiltonian circuit is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity wi.

Its negative resolution by Leonhard Euler in  laid the foundations of graph theory and prefigured the idea of topology. The problem was to devise a walk through the city that would cross each of those bridges once and only once. By way of specifying the logical task unambiguously, solutions involving either reaching an island or mainland bank other than via one of the bridges, or accessing any bridge without crossing to its other end are explicitly unacceptable. Euler proved that the problem has no solution.

## Eulerian Graphs and Related Topics, Volume 1 - 1st Edition

The difficulty he faced w. A multigraph with vertices labeled by degree In graph theory, the degree or valency of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. Leonhard Euler — In mathematics and physics, a large number of topics is named in honor of Swiss mathematician Leonhard Euler — , who made many important discoveries and innovations.

Many of these items named after Euler include their own unique function, equation, formula, identity, number single or sequence , or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.

Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler. Look up unicursal in Wiktionary, the free dictionary. Unicursal may refer to: Labyrinth - a unicursal maze Unicursal hexagram - a star polygon Eulerian path - a sequential set of edges within a graph that reach all nodes unicursal curve - A curve which is birationally equivalent to a line.

This is a list of graph theory topics, by Wikipedia page. Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes. In contrast to genetics, which refers to the study of individual genes and their roles in inheritance, genomics aims at the collective characterization and quantification of all of an organism's genes, their interrelations and influence on the organism.

In turn, proteins make up body structures such as organs and tissues as well as control chemical reactions and carry signals between cells. Genomics also involves the sequencing and analysis of genomes through uses of high throughput DNA sequencing and bioinformatics to assemble and analyze the function and structure of entire genomes. This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges. Prime symbol ' The prime symbol is often used to modify notation for graph invariants so that it applies to the line graph instead of the given graph.

A achromatic The achromatic number of a graph is the maximum number of colors in a complete coloring. A graph is acyclic if it has no cycles. An undirected acyclic graph is the same thing as a forest. Directed acyclic graphs are less often called acyclic directed graphs. An acyclic coloring of an undirecte. A simple rendition of the Five-room puzzle This classical, popular puzzle involves a large rectangle divided into five "rooms".

The objective of the puzzle is to cross each "wall" of the diagram with a continuous line only once. The numbers denote the number of edges connected to each vertex. Vertices with an odd number of edges are shaded orange. Because there is more than one pair of vertices with an odd number of edges, the resulting multigraph does not contain an Eulerian path nor an Eulerian. In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols.

It has mn vertices, consisting of all possible length-n sequences of the given symbols; the same symbol may appear multiple times in a sequence.

## Eulerian path

In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed oriented graphs. The name is an acronym of the names of people who discovered it: de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte. An Eulerian circuit is a directed closed path which visits each edge exactly once.

In , Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg v. In general there are many sequences for a particular n and k but in this example it is unique, up to cycling.

In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring i. Such a sequence is denoted by B k, n and has length kn, which is also the number of distinct substrings of length n on A; de Bruijn sequences are therefore optimally short. There are k! The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn.

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According to him, the existence of de Bruijn sequences for each order together with the above properties were first proved, for the case of alphabets with two elements, by Camille Flye Sainte-Marie i. Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot The travelling salesman problem TSP asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city? The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.

In the theory of computational complexity, the decision version of the TSP where, given a length L, the task is to decide whether the graph has any tour shorter than L belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially but no more than exponentially with the number of cities. The problem was first formulated in and is one of the mos.

An expanse of driftwood along the northern coast of Washington state. Stokes drift — besides e. Ekman drift and geostrophic currents — is one of the relevant processes in the transport of marine debris. Click here for an animation 4. Description also of the animation : The red circles are the present positions of massless particles, moving with the flow velocity. The light-blue line gives the path of these particles, and the light-blue circles the particle position after each wave period. The white dots are fluid particles, also followed in time. In the case shown here, the mean Eulerian horizontal velocity below the wave trough is zero.

Observe that the wave period, experienced by a fluid particle near the free surface, is different from the wave period at a fixed horizontal position as indicated by the light-blue circles. This is due to the Doppler shift. Stokes drift in shallow water waves, with a wa. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.

The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circl. Robot in a wooden maze There are a number of different maze solving algorithms, that is, automated methods for the solving of mazes.

Mazes containing no loops are known as "simply connected", or "perfect" mazes, and are equivalent to a tree in graph theory. Thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.

It is simply to proceed following the current passage until a junction is reached, and then to make a random decision about the. In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices at least 3 connected in a closed chain. The cycle graph with n vertices is called C. Terminology There are many synonyms for "cycle graph".

These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used. The term n-cycle is sometimes used in other settings. Properties A cycle graph is: 2-edge colorable, if and only if it has an even number of vertices 2-regular 2-vertex colorable, if and only if it has an even number of verti.

In graph theory, a branch of mathematics, the binary cycle space of an undirected graph is the set of its Eulerian subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings.

Definitions Graph theory A spanning subgraph of a given graph G may be defined from any subset S of the edges of G. The subgraph has the same set of vertices as G itself this is the meaning of the word "spanning" but has the elements of S as its edges. Thus, a graph G with m edges has 2m spanning subgraphs, including G itself as well as the empty graph on the same set of vertices as G.

The collection of all spanning subgraphs of a graph G forms the edge space of G. In graph theory, a branch of mathematics and computer science, the Chinese postman problem CPP , postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of an connected undirected graph.

When the graph has an Eulerian circuit a closed walk that covers every edge once , that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate or the subset of edges with the minimum possible total weight so that the resulting multigraph does have an Eulerian circuit. Goldman or Jack Edmonds, both of whom were at the U. National Bureau of Standards at the time. The symmetric difference of two cycles is an Eulerian subgraph In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph.

That is, it is a minimal set of cycles that allows every Eulerian subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. If such a cycle exists, the graph is called Eulerian or unicursal. For finite connected graphs the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the weaker sense if and only if each connected component has an Eulerian cycle.

For directed graphs , "path" has to be replaced with directed path and "cycle" with directed cycle.

EULERIAN GRAPH

The definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs as well. An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v , the indegree of v equals the outdegree of v.

Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour. Fleury's algorithm is an elegant but inefficient algorithm that dates to The algorithm starts at a vertex of odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge left at the current vertex.

It then moves to the other endpoint of that edge and deletes the edge. At the end of the algorithm there are no edges left, and the sequence from which the edges were chosen forms an Eulerian cycle if the graph has no vertices of odd degree, or an Eulerian trail if there are exactly two vertices of odd degree. While the graph traversal in Fleury's algorithm is linear in the number of edges, i. O E , we also need to factor in the complexity of detecting bridges.

If we are to re-run Tarjan 's linear time bridge-finding algorithm after the removal of every edge, Fleury's algorithm will have a time complexity of O E 2. Hierholzer 's paper provides a different method for finding Euler cycles that is more efficient than Fleury's algorithm:. This algorithm may also be implemented with a queue. Because it is only possible to get stuck when the queue represents a closed tour, one should rotate the queue remove an element from the head and add it to the tail until unstuck, and continue until all edges are accounted for.

The formula states that the number of Eulerian circuits in a digraph is the product of certain degree factorials and the number of rooted arborescences. The latter can be computed as a determinant , by the matrix tree theorem , giving a polynomial time algorithm. BEST theorem is first stated in this form in a "note added in proof" to the Aardenne-Ehrenfest and de Bruijn paper