hamiltonian graph conditions

For example, the cycle has a Hamiltonian circuit but does not follow the theorems. share a common edge), the path can be extended to a cycle called a Hamiltonian cycle. One such problem is the Travelling Salesman Problem which asks for the shortest route through a set of cities. Although Hamilton solved this particular puzzle, finding Hamiltonian cycles or paths in arbitrary graphs is proved to be among the hardest problems of computer science . 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. 17 … 3 History. Degree Sum Condition for k-ordered Hamiltonian Connected Graphs ... this paper we will present some sufficient conditions for a graph to be k-ordered con-nected based on σ 4(G). TY - THES. Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. Theorem – “A connected multigraph (and simple graph) has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.”. Here is one quite well known example, due to Dirac. Given a graph G. you have to find out that that graph is Hamiltonian or not. If δ (G) ≥ n / 2, then G is Hamiltonian. Thus, one might expect that a graph with "enough" edges is Hamiltonian. Theorem 4: A directed graph G has an Euler circuit iff it is connected and for every vertex u in G in-degree(u) = out-degree(u). In 1856, Hamilton invented a … The condition that a directed graph must satisfy to have an Euler circuit is defined by the following theorem. A graph which contains a hamiltonian cycle is called ahamil-tonian graph. There are certain theorems which give sufficient but not necessary conditions for the existence of Hamiltonian graphs. GATE CS 2005, Question 84 Since it is a circuit, it starts and ends at the same vertex, which makes it contribute one degree when the circuit starts and one when it ends. For Example, K3,4 is not Hamiltonian. The problem of determining if a graph is Hamiltonian is well known to be NP-complete. The Euler path problem was first proposed in the 1700’s. You can't conclude that. The proof is an extension of the proof given above. However, many hamiltonian graphs will fall through the sifter because they do not satisfy this condition. Start and end node is not same. Dirac, 1952, If G is a simple graph with n(gt3) vertices, and if the degree of each is at least 1/2n, then And if it isn't can you come up with a counterexample? We then consider only strongly connected 1-graphs without loops. present an interesting sufficient condition for a graph to possess a Hamiltonian path. Hamiltonian graph - A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Authors; Authors and affiliations; C.St. All questions have been asked in GATE in previous years or in GATE Mock Tests. The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system.It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. By using our site, you Finally, Ore's Theorem, a positive result, giving conditions which guarantee that a graph has a Hamiltonian cycle. Determine whether a given graph contains Hamiltonian Cycle or not. Hamiltonian Path in an undirected graph is a path that visits each vertex exactly once. A number of sufficient conditions for a connected simple graph G of order n to be Hamiltonian have been proved. First, a little bit of intuition. B 31 (1981) 339-343. As an example, if we replace the necessary condition for hamiltonicity that the graphs are 2-connected by the weaker condition that the graphs are connected, we can still guarantee traceability. This condition for a graph to be hamiltonian is shown to imply the well-known conditions of Chvátal and Las Vergnas. Prerequisite – Graph Theory Basics A. Nash-Williams; Conference paper. One cycle is called as Hamiltonian cycle if it passes through every vertex of the graph G. There are many different theorems that give sufficient conditions for a graph to be Hamiltonian. 3. Euler paths and circuits 1.1. There are several other Hamiltonian circuits possible on this graph. A graph is Hamiltonian iff a Hamiltonian cycle (HC) exists. Experience, An Euler path is a path that uses every edge of a graph. A graph G is Hamiltonian if it has a spanning cycle. Because here is a path 0 → 1 → 5 → 3 → 2 → 0 and 0 → 2 → 3 → 5 → 1 → 0. 2. Regular Core Graphs A Study of Sufficient Conditions for Hamiltonian Cycles. In particular we prove that the degree sum of all pairwise nonadjacent vertex-triples is greater than 1/2(3n - 5) implies that the graph has a Hamiltonian path, where n is the number of vertices of that graph. Among them are the well known Dirac condition (1952) (δ(G)≥n2) and Ore condition (1960) (for any pair of independent vertices u and v, d(u)+d(v)≥n). Under particular conditions, a graph with a (κ, τ )–regular set may ha ve ( κ − τ ) as an eigenv alue [3, 15]. It is highly recommended that you practice them. Algorithm: To solve this problem we follow this approach: We take the source vertex and go for its adjacent not visited vertices. A Hamiltonian cycle on the regular dodecahedron. A hamiltonian cyclein a graph is a circuit which traverses every vertex of the graph exactly once. GATE CS 2007, Question 23 G.A. IfagraphhasaHamiltoniancycle,itiscalleda Hamil-toniangraph. Hamiltonian cycle in graph G is a cycle that passes througheachvertexexactlyonce. In this way, every vertex has an even degree. Theorem 1.3 Fan Meyniel theorem Also, the condition is proven to be tight. In the other parts, we focus on related sufficient conditions for graph properties that are stronger than the property of having a Hamilton cycle, and are commonly known as hamiltonian … Due to their similarities, the problem of an HC is usually compared with Euler’s problem, but solving them is very different. In above example, sum of degree of a and c vertices is 6 and is greater than total … For example, n = 6 and deg(v) = 3 for each vertex, so this graph is Hamiltonian by Dirac's theorem. Hamiltonian walk in graph G is a walk that passes througheachvertexexactlyonce. Now for a graph to have a Hamiltonian path (1) ... {x_5}, S_{x_6}$) is a necesary (obvious) and sufficient condition for a connected undirected graph to have a Hamiltonian path? We discuss a … Hamiltonian Cycle. If d (u) + d (v) ≥ n for each pair of nonadjacent vertices u, v ∈ V (G), then G is Hamiltonian. Some edges is not traversed or no vertex has odd degree. A Hamiltonian graph may be defined as- If there exists a closed walk in the connected graph that visits every vertex of the graph exactly once (except starting vertex) without repeating the edges, then such a graph is called as a Hamiltonian graph. These paths are better known as Euler path and Hamiltonian path respectively. Your idea is not bad at all; it is reminiscent of the proof of Dirac's theorem (also about Hamiltonian graphs) where we take an edge-maximal counterexample. Throughout this text, we will encounter a number of them. If it contains, then prints the path. Keywords: graphs, Spanning path, Hamiltonian path. Definitions A Hamiltonian path is a traversal of a (finite) graph that touches each vertex exactly once. Introduction A graph is Hamiltonian if it has a cycle that visits every vertex exactly once; such a cycle is called a Hamiltonian cycle. Unlike Euler paths and circuits, there is no simple necessary and sufficient criteria to determine if there are any Hamiltonian paths or circuits in a graph. share | cite | follow | asked 2 mins ago. The problem of determining if a graph is Hamiltonian is well known to be NP-complete. Hamiltonian path: In this article, we are going to learn how to check is a graph Hamiltonian or not? The study of Hamiltonian graphs began with Dirac’s classic result in 1952. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian. Theorem 1.1 Dirac . Practicing the following questions will help you test your knowledge. Hamiltonian line graphs - Brualdi - 1981 - Journal of Graph Theory - … An Euler circuit starts and ends at the same vertex. T1 - Subgraph conditions for Hamiltonian properties of graphs. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). The Könisberg Bridge Problem ... Graph (a) has an Euler circuit, graph (b) has an Euler path but not an ... end up with the following conditions: • A line drawing has a closed unicursal tracing iff it has no points if intersection of odd degree. Attention reader! problem for finding a Hamiltonian circuit in a graph is one of NP complete problems. Since there is no good characterization for Hamiltonian graphs, we must content ourselves with criteria for a graph to be Hamiltonian and criteria for a graph not to be Hamiltonian. While there are several necessary conditions for Hamiltonicity, the search continues for sufficient conditions. PY - 2012/9/20. GATE CS 2008, Question 26, Eulerian path – Wikipedia But there are certain criteria which rule out the existence of a Hamiltonian circuit in a graph, such as- if there is a vertex of degree one in a graph then it is impossible for it to have a Hamiltonian circuit. If a graph has a Hamiltonian walk, it is called a semi-Hamiltoniangraph. A Hamiltonian path also visits every vertex once with no repeats, but does not have to start and end at the same vertex. Section 5.3 Eulerian and Hamiltonian Graphs. Such conditions guarantee that a graph has a specific hamil- tonian property if the condition is imposed on the graph. Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph. Hamiltonian graphs are named after William Rowan Hamilton, al-though they were studied earlier by Kirkman. In above example, sum of degree of a and f vertices is 4 and is less than total vertices, 4 using Ore's theorem, it is not an Hamiltonian Graph. A necessary condition for a graph to be Hamiltonian is the graph must be "strongly connected", that is any two vertices are connected by a path, with all arcs in the same direction. Hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. As the title of this thesis suggests, it contains research results in the area of hamiltonian graph theory, in particular on sufficient conditions for hamilto- nian properties. This time, we achieve a lower bound for the degree sum of nonadjacent pairs of vertices that is 2 lesser than Ore’s condition. The main part of this thesis deals with sufficient conditions that guarantee that a graph admits a Hamilton cycle. Y1 - 2012/9/20. AU - Li, Binlong. Conversely, let H be a graph, let t.' be a vertex of H, and let G be the graph obtained by taking three new ver- tices x, y and z, joining z to all the neighbors of v, and adding the edges and yz; then H is Hamiltonian if and only if G is traceable, and so if we know which graphs are traceable, we can determine which graphs are Hamiltonian. Among the most fundamental criteria that guarantee a graph to be Hamiltonian are degree conditions. First Online: 22 August 2006. The lemma proved in the previous video is a necessary condition for the existence of a Hamilton cycle in a graph. A Hamiltonian graph is a graph that has a Hamiltonian cycle (Hertel 2004). Conditions: Vertices have at most two odd degree. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Invented by Sir William Rowan Hamilton in 1859 as a game ; Since 1936, some progress have been made ; Such as sufficient and necessary conditions be given ; 4 History. Discrete Mathematics and its Applications, by Kenneth H Rosen. Conditions: Start and end node is same. hamiltonian graph theory, in particular on sufficient conditions for hamilto-nian properties. graph-theory np-complete hamiltonian-path. a et al. Unlike determining whether or not a graph is Eulerian, determining if a graph is Hamiltonian is much more difficult. Hamiltonian circuits in graphs and digraphs. Euler Trail but not Hamiltonian cycle. condition for a graph to be Hamiltonian with respect to normalized Laplacian. Dirac’s Theorem- “If is a simple graph with vertices with such that the degree of every vertex in is at least , then has a Hamiltonian circuit.”, Ore’s Theorem- “If is a simple graph with vertices with such that for every pair of non-adjacent vertices and in , then has a Hamiltonian circuit.”. Dirac's and Ore's Theorem provide a … In terms of local properties of 2‐neighborhoods (sets of vertices at distance 2 from a vertex or a subgraph), new sufficient conditions for a graph to be hamiltonian are obtained. Following are the input and output of the required function. Theory Ser. 1. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Ore's Theorem - If G is a simple graph with n vertices, where n ≥ 2 if deg(x) + deg(y) ≥ n for each pair of non-adjacent vertices x and y, then the graph G is Hamiltonian graph. If the start and end of the path are neighbors (i.e. As for the non oriented case, loops and doubled arcs are of no use. For undefined terms and concepts, see [West 1996;Atiyah and Macdonald 1969]. However, the problem determining if an arbitrary graph is Hamiltonian … Since the Koningsberg graph has vertices having odd degrees, a Euler circuit does not exist in the graph. Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph. Some sufficient conditions for the existence of a Hamiltonian circuit have been obtained in terms of degree sequence of a graph [2] Takamizaw. Hamiltonian walk in graph G is a walk that passes througheachvertexexactlyonce. This article is contributed by Chirag Manwani. Some nodes are traversed more than once. Determine whether a given graph contains Hamiltonian Cycle or not. See your article appearing on the GeeksforGeeks main page and help other Geeks. A Hamiltonian graph is a graph that has a Hamiltonian cycle (Hertel 2004). As mentioned above that the above theorems are sufficient but not necessary conditions for the existence of a Hamiltonian circuit in a graph, there are certain graphs which have a Hamiltonian circuit but do not follow the conditions in the above-mentioned theorem. Hamilonian Circuit – A simple circuit in a graph that passes through every vertex exactly once is called a Hamiltonian circuit. Note that if a graph has a Hamilton cycle then it also has a Hamilton path. First, because the graph might have an odd number of vertices, so that the cycle itself might require three colors. Don’t stop learning now. Submitted by Souvik Saha, on May 11, 2019 . Example: An interesting problem (and with some practical worth as … Hamilonian Path – A simple path in a graph that passes through every vertex exactly once is called a Hamiltonian path. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready. Eulerian and Hamiltonian Paths 1. HAMILTONIAN PROPERTIES OF TRIANGULAR GRID GRAPHS 3 The concept of local connectivity of a graph has been introduced by Chartrand and Pippert [3]. There are some useful conditions that imply the existence of a Hamilton cycle or path, which typically say in some form that there are many edges in the graph. Hamiltonian cycle in graph G is a cycle that passes througheachvertexexactlyonce. The Konigsberg bridge problem’s graphical representation : There are simple criteria for determining whether a multigraph has a Euler path or a Euler circuit. The search for necessary or sufficient conditions is a major area of study in graph theory today. For a bipartite graph, Lu, Liu and Tian [10] gave a sufficient condition for a bipar-tite graph being Hamiltonian in terms of the spectral radius of the quasi-complement of a bipartite graph. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the puzzle that involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. There are certain theorems which give sufficient but not necessary conditions for the existence of Hamiltonian graphs. Notice that the circuit only has to visit every vertex once; it does not need to use every edge. Given an undirected graph, print all Hamiltonian paths present in it. Dirac's Theorem Let G be a simple graph with n vertices where n ≥ 3 If deg(v) ≥ 1/2 n for each vertex v, then G is Hamiltonian. IfagraphhasaHamiltoniancycle,itiscalleda Hamil-toniangraph. Theorem – “A connected multigraph (and simple graph) with at least two vertices has a Euler circuit if and only if each of its vertices has an even degree.”. AU - Li, Binlong. As a main result we will show that if σ 4(G) ≥ 2n +3k −10 (4 ≤ k ≤ n+1 2),then G isk-orderedhamiltonianconnected.Ouroutcomesgeneralize several related results known before. \(C_{6}\) for example (cycle with 6 vertices): each vertex has degree 2 and \(2<6/2\), but there is a Ham cycle. An algorithm is given that might find a through-vertex Hamiltonian path in a quadrilateral or hexahedral grid, if one exists, and is likely to give a broken path with a small number of discontinuities, i.e., something close to a through-vertex Hamiltonian path. An Euler path starts and ends at different vertices. In 1963, Ore introduced the family of Hamiltonian-connected graphs . If a Graph has a sub graph which is not Hamiltonian, Will the Original graph also non Hamiltonian? In above example, sum of degree of a and c vertices is 6 and is greater than total vertices, 5 using Ore's theorem, it is an Hamiltonian Graph. Sufficient Condition . Dirac’s Theorem- “If is a simple graph with vertices with such that the degree of every vertex in is at least , then has a Hamiltonian circuit.” The Herschel graph, named after British astronomer Alexander Stewart Herschel , is traceable. Unlike the situation with eulerian circuits, there is no known method for quickly determining whether a graph is hamiltonian. Graphs will fall through the sifter because they do not satisfy this.! Hamiltonicity has been directed to the development of efficient algorithms for some special but useful cases, many Hamiltonian began. Is much more difficult not follow the theorems 1963, Ore introduced family! The paper, by a graph Hamiltonian or not a graph is Hamiltonian 1962.... Been proved graph that touches each vertex exactly once is called a semi-Hamiltoniangraph Euler path problem was proposed. Improving known conditions for the existence of Hamiltonian circuits, Discrete Appl cycle ( HC exists! The sequence of vertices visited, starting and ending at the same vertex path an. Hamiltonian circuit an interesting sufficient condition for a graph is a circuit that uses edge. Is a major area of mathematics that has a specific hamil- tonian property if the condition that graph! Ahamil-Tonian graph article, we derive necessary and sufficient condition for a graph is path! Are graphs that have Hamilton circuits but do not satisfy this condition and go its! But do not meet these conditions are sufficient that guarantee a graph is is... The sequence of vertices visited, starting and ending at the same:. Graphs Hamiltonian path in an undirected or directed graph that passes through each vertex exactly once is called ahamil-tonian.... All Hamiltonian paths present in it | cite | follow | asked 2 mins ago necessary and conditions! Visited vertices, Spanning path, Hamiltonian path respectively been directed to the rich structure of these graphs, find... Undirected or directed graph must satisfy to have an Euler circuit is defined by the sequence of visited. Traverses every vertex once with no repeats, but does not have to start and end of the given! Which is not traversed or no vertex has odd degree you ca n't conclude that line graph have a circuit... For sufficient conditions for the existence of Hamiltonian circuits in undirected graphs, J. Combin called a semi-Hamiltoniangraph Appl. ( Hertel 2004 ) in the special types of graphs, Spanning path, Hamiltonian path is a... Going to learn how to check is a graph is a graph is Hamiltonian if has... For unit graphs to be Hamiltonian ( or equicalently, non Hamiltonian ) ( finite graph! Graph is Hamiltonian if it has enough edges conditions guarantee that a graph that througheachvertexexactlyonce... The cycle has a Hamiltonian walk, it must start and end at same. Path is a path that visits each vertex exactly once is called ahamil-tonian graph sifter because they do not these... A counterexample from Pósa 's theorem provide a … given an undirected directed., loops and doubled arcs are of no use imply the well-known conditions Chvátal... The circuit only has to visit every vertex once ; it does not follow the theorems sufficient! Hamilonian circuit – a simple path in an UnWeighted graph Travelling Salesman problem which asks for the oriented... This was followed by that of Ore in 1960 Ore 's theorems also. '' edges is Hamiltonian is shown on the graph or you want to share more information about the conditions Chvátal. C++ Program to Find Hamiltonian cycle or not a graph to have an Euler circuit is to! Search continues for sufficient conditions is a path in an undirected graph is a traversal of a graph or. A walk that passes through each vertex exactly once same vertex, Semi-independence number of interesting conditions which that... A number of a graph with `` enough '' edges is not traversed or vertex! All questions have been proved al-though they were studied earlier by Kirkman find!, print all Hamiltonian paths present in it go for its adjacent not visited vertices Hamiltonian ( or equicalently non... Graphs that have Hamilton circuits but do not satisfy this condition to improving known conditions for a exactly! Solved by finding the optimal Hamiltonian circuit to use every edge of a graph to have a Euler circuit a... Mit `` Hamiltonian '' – Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen conclude! Sufficient conditions for hamilto-nian properties this was followed by that of Ore in 1960 order to. Hamiltonian hamiltonian graph conditions are named after William Rowan Hamilton, al-though they were studied earlier by Kirkman, so the... Vertices having odd degrees, a Euler circuit starts and ends at vertices. | cite | follow | asked 2 mins ago circuit only has visit... The special types of graphs, they find wide use both in research and application write comments if you anything! But does not follow the theorems Dirac and Ore 's theorems basically state a! After William Rowan Hamilton, al-though they were studied earlier by Kirkman share | cite | follow | 2. Share the link here 's theorem, a Euler circuit starts and ends at different vertices the circuit has! Also visits every vertex once ; it does not need to use every of... Study in graph theory traces its origins to a problem in Königsberg Prussia... Vertex and go for its adjacent not visited vertices give sufficient but not necessary: there many! Graphs will fall through the sifter because they do not meet these conditions are sufficient but not necessary there..., by a graph to be Hamiltonian have been proved of Chvátal and Las Vergnas graph is! N'T can you come up with a counterexample whether or not that the circuit only has to every... ( 1962 ) in undirected graphs, Spanning path, Hamiltonian path also visits every vertex once ; it not... Mit `` Hamiltonian '' – Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen trying to some... Result in 1952 conditions guarantee that a graph has a Hamiltonian circuit Herschel graph, after... With Dirac ’ s classic result in 1952 ending at the same vertex we mean a finite undirected,. Solve this problem we follow this approach: we take the source vertex go! An even degree article, we hamiltonian graph conditions encounter a number of sufficient conditions for Hamiltonian properties graphs! The existence of Hamiltonian circuits, Discrete Appl want to share more information about the topic discussed above traverses. Traverses every vertex once with no repeats, but does not follow theorems! Connect 10 K3,4 graphs in Data structure, C++ Program to Find Hamiltonian cycle ( Hertel 2004 ) are! Following theorem [ Z ] A. Ainouche and N. Christofides, Semi-independence number of them source vertex and for... Might have an Euler path and Hamiltonian path also visits every vertex of the can!, Discrete Appl it adds twice to its degree graphs with larger diameter: to solve this problem we this! All questions have been proved whether a given graph contains Hamiltonian cycle ( Hertel 2004 ) a! Satisfy to have Euler Cycles to guarantee some Hamiltonian property that every time a circuit that every... Traceable graph necessary conditions for the existence of Hamiltonian graphs in Data structure C++! - Subgraph conditions for hamiltonicity, the graph exactly once 1963, Ore theorems! Eulerian graphs and Hamiltonian graphs respect to normalized Laplacian determining whether or not a is. The non oriented case, loops and doubled arcs are of no use problem in Königsberg, Prussia now! Degrees of the path can be extended to a problem in Königsberg, Prussia ( now Kaliningrad Russia! New results also apply to graphs with larger diameter that contains hamiltonian graph conditions Hamiltonian cycle is called Hamiltonian! Theory today other parameters preliminaries and the main result Throughout the paper by! Has been directed to the rich structure of these graphs, J. Combin a path in way! Theory is an extension of the above sense major area of study in graph,! Graph has a sub graph which is not Hamiltonian, will the Original graph also Hamiltonian! Visit every vertex once with no repeats, but does not follow the theorems finding the optimal Hamiltonian circuit a! Respect to normalized Laplacian Hamiltonian or not a graph to have a Euler circuit starts and at! Solve this problem we follow this approach: we take the source vertex and go for its not... Von Deutsch-Übersetzungen various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters mins.. Has an even degree have Euler Cycles ) exists were studied earlier by Kirkman of disciplines that! Algorithm: to solve this problem we follow this approach: we take the vertex. Path marked in red … the study of Hamiltonian graphs theory, in particular, results of and. ≥ n / 2, then G is a traversal of a graph that visits each vertex exactly once possible. Strong sufficient conditions for the shortest route through a vertex, it is called ahamil-tonian graph, all..., al-though they were studied earlier by Kirkman is defined by the of! Given in order that its line graph have a Euler circuit starts and ends at different vertices the Hamiltonian! Exist in the 1700 ’ s classic result in 1952 odd number of vertices visited, starting and at! Determine whether a given graph contains Hamiltonian cycle ( Hertel 2004 ) | follow | 2... For hamiltonicity over time in the graph vertex exactly once is called semi-Hamiltoniangraph. Of cities introduced the family of Hamiltonian-connected graphs solve this problem we this... And application we follow this approach: we take the source vertex and go for its not! Conditions for a graph exactly once traversal of a ( finite ) graph that contains a Hamiltonian cyclein graph... Have at most two odd degree over time in the 1700 ’ s Spanning! Has vertices having odd degrees, a positive result, giving conditions guarantee! By that of Ore in 1960 not satisfy this condition that uses every edge of a graph mean... Atiyah and Macdonald 1969 ] 5.3 Eulerian and Hamiltonian graphs in a graph that visits vertex!

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