As the graph is the complete bipartite graph, we can count the number of cycle as : Choose an initial set; On the first set, you have $n$ choices for the first vertex; On the second again $n$ choices; Then $n-1$ choices; and so on $\ldots$ Therefore we count H=2(n!)(n!) Hamiltonian cycles We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 2 1−d d! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2) g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs Input : N = 6 Output : Hamiltonian cycles = 60 Input : N = 4 Output : Hamiltonian cycles = 3 Recommended: Please try your approach on {IDE} first, before moving on to the solution. Explanation

The paper is organized as follows. A method for counting **Hamiltonian** **cycles** is developed in Section 2. In Section 3, the n-cubeisusedasanexample,and the **number** **of** **Hamiltonian** **cycles** **in** the 6-cube is determined. The **number** **of** **Hamiltonian** **cycles** up to the symmetry of the 6-cube is further obtained usin Which complete bipartite graphs are Hamiltonian? We'll prove the answer to that question in today's graph theory lesson!A little bit of messing around with c.. The number of Hamiltonian cycles in the complete bipartite graphCounting Hamiltonian cycles in a... Hackerrank All Women's Codesprint 2019: Name the Product PTIJ: Which Dr. Seuss books should one obtain? Why are there no stars visible in cislunar space? How to test the sharpness of a knife? When should a starting writer get his own webpage? How to balance a monster modification (zombie.

In essence, it claimed the number of Hamiltonian cycles in a bipartite cubic (3 regular) graph is even. This is very similar to problem 5 of Bulgarian NMO, 2020, where a 4 regular graph was given and something about the parity of its hamiltonian decompositions was claimed ** You have made an error when you have calculated the total number of cycles**. A Hamiltonian cycle must include all the edges. k4 has only 3 such cycles and in total it has 5 cycles, so the formula is correct. - Anubhav Apr 19 '13 at 17:3 In the following, G(a, b, k) is a simple bipartite graph with bipartition (A, B), where JA I = a > 2, 1 B I = b > k, and each vertex of A has degree at least k. We shall prove two results on the existence of certain cycles in G(a, b, k). The first result was stated as a conjecture by Sheehan [4]. THEOREM 1. If a graph G(a, b, k) satisfies a G k and then it contains a cycle of length 2a. Theorem 1 is best possible in the following sense. Consider a separable graph G(a, 2k - 1, k.

** As A is a complete bipartite graph**, there are k edge-disjoint hamiltonian cycles in (A X ′ × A Y) G, and we let x 1 y 1, , x k y k be independent edges of (A X ′ × A Y) G such that x i y i is an edge of the i th hamiltonian cycle The number of different Hamiltonian cycles in a complete undirected graph on n vertices is (n − 1)! / 2 and in a complete directed graph on n vertices is (n − 1)!. These counts assume that cycles that are the same apart from their starting point are not counted separately. Bondy-Chvátal theore The complete bipartite graph K n;n is Hamiltonian, for all n 2. Proof. K n;n is a simple graph on 2nvertices. So for n 2, we have that K n;n has at least 3 vertices. Under this restriction, a su cient condition for Hamiltonicity is that the degree of every vertex is greater than or equal to half the number of vertices. As the degree of each vertex in K n;n is n(= 2n=2), we have our desired.

It's not too hard to check that if $|V|$ is odd, $G$ has a Hamiltonian cycle if and only if $\tilde{G}$ is. If $|V|$ is even, just add one new vertex of degree 2 with neighbors that have an edge between them. So far, we've seen that Hamiltonian Cycle in bipartite graphs is hard. To reduce this to Hamiltonian Path, just continue in the way you normally would to reduce Hamiltonian Cycle to Hamiltonian Path Hamilton Cycles in Bipartite Graphs Theorem If a bipartite graph has a Hamilton cycle, then it must have an even number vertices. Theorem K m;n has a Hamilton cycle if and only if m = n 2. 11/2 Lu, X. Hamiltonian cycles in bipartite graphs. Combinatorica 15, 247-254 (1995). https://doi.org/10.1007/BF01200758. Download citation. Received: 05 January 1993. Revised: 10 May 1993. Issue Date: June 1995. DOI: https://doi.org/10.1007/BF0120075

- imum degree d has at least 2 1−d d! Hamiltonian cycles, and every bipartite Hamiltonian graph of
- every bipartite graph with exactly n=2 vertices in each partite set is Hamiltonian when any two nonadjacent vertices taken from di erent partite sets have degree-sum at least n=2+1. All the mentioned su cient conditions are sharp in the sense that one cannot make thei
- Hamiltonian cycle on grid graphs containing holes is NP-complete by reduction from Hamiltonian cycle on planar undirected bipartite max-degree-3 graphs [IPS82]. The reduction rst embeds the input bipartite graph on a grid graph using the grid graph's inherent 2-coloring. Any parity violations can be sidestepped by scaling the grid graph by a factor of 3, allowing wiggles to be added as required
- about when the number of Hamiltonian cycles is greater than one. Another interesting class of graphs are the bipartite graphs. In general these are not Hamiltonian but there is a famous conjecture due to Barnette that suggests that 3-connected cubic bipartite planar graphs are Hamiltonian. In our final two sections we consider this along with another ope
- ing whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph. Both problems are NP-complete. The Hamiltonian cycle problem is a special case of the travelling salesman problem, obtained by setting the distance between two cities to one if they are adjacent and two otherwise, and verifying that the total distance travelled is equal to n
- Theorem 1.5. If G = (X,Y) is a balanced bipartite graph of order 2n, with n ≥ 128k2 such σ2 2 (G) ≥ n+2k −1, then G has k edge-disjoint hamiltonian cycles. 2 Veneering Numbers and k-Extendibility To prove our main theorem, we need some results on path-systems in bipartite graphs
- ant of a given matrix can be ex- panded by its principal

Every cubic 3-connected bipartite planar graph is Hamiltonian. Conjecture 2 (Jaeger). Every cubic cyclically 4-edge connected graph G has a cycle C such that G-V(C) is acyclic. Conjecture 3. Abstract We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors Complete for general graphs [3]. The problem remains NP-complete [3] (1) Taking up the theme regarding the number of Hamiltonian cycles in a Hamiltonian graph, Thomassen [11] has provided a condition for a bipartite Hamiltonian graph to have a second Hamiltonian cycle. Theorem 5(see [11]). Let G be a Hamiltonian graph with bipartition A, B and let H be a Hamiltonian cycle in G. If every.

Let G be a balanced bipartite graph with bipartite sets X, Y. We say that G is Hamilton-biconnected if there is a Hamilton path connecting any vertex in X and any vertex in Y . We define the bipartite independent number \(\alpha ^o_B(G)\) to be the maximum integer \(\alpha \) such that for any integer partition \(\alpha =s+t\) , G has an independent set formed by s vertices in X and t vertices in Y Abstract We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors. Alternating hamiltonian cycles in two colored complete bipartite graphs - Chetwynd - 1992 - Journal of Graph Theory - Wiley Online Librar You can try some Monte Carlo simulations to guess at the number: suppose you choose 10 vertices on one side of the bipartite graph, and 20 edges coming from these vertices; how many Hamiltonian cycles can you build using those 20 edges as a base. If the number is 0 or small then you might have a feasible enumeration; if it is large then you might try estimating the logarithm of your number.

Hamiltonian Cycles in Products of Graphs - Volume 17 Issue 5. To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account ** Volume 111, Number 1, January 1991 ENUMERATION OF HAMILTONIAN CYCLES AND PATHS IN A GRAPH C**. J. LIU (Communicated by Andrew Odlyzko) Abstract. First, we show that the determinant of a given matrix can be ex- panded by its principal minors together with a set of arbitrary parameters. The enumeration of Hamiltonian cycles and paths in a graph is then carried out by an algebraic method. Three. converted to one and only one Hamiltonian cycle in G+ ; therefore we have Theorem 8. Let H be the number of Hamiltonian paths of a given graph. Then (40) H P in G = H c in G . Application. We consider here the applications of Theorem 6 and 7 to a com-plete multipartite graph Kn ni'np . It can be shown that the number of spannin 1.2. The number of Hamiltonian cycles Let h(n;k) be the minimum number ofHamiltonian cycles over all labelled graphs ofordern and minimum degree k. At one extreme h(n;n − 1)=(n − 1)! At the other extreme, since the bipartite graph K((n+1)=2 ; (n−1)=2) does not have any Hamiltonian cycles, h(n;k)=0 for all k6 (n − 1)=2. A classical result ofDirac [6 Note! The problem of determining the number of Hamiltonian cycles in a graph is #P-complete and determining whether it is > 0 is NP-complete. Possible variants and generalizations: I Consider nonbipartite graphs I Consider directed graphs I Count Hamiltonian paths I Count (perfect) matchings I Snake-in-the-box (longest induced path

** Taking up the theme regarding the number of Hamiltonian cycles in a Hamiltonian graph, Thomassen [11] has provided a condition for a bipartite Hamiltonian graph to have a second Hamiltonian cycle**. Theorem 5(see [11]). Let G be a Hamiltonian graph with bipartition A, B and let H be a Hamiltonian cycle in G. I the number of Hamiltonian cycles in graphs Pm Pn and Cm Pn [16,17]. Earlier, Saburo developed a ﬁeld theoretic approximation of the number of Hamiltonian cycles in graphs Cm Cn in [18] as well as in planar random lattices [19]. Fireze et al. considered generating and counting Hamiltonian cycles in random regular graphs [20]. Given these results, we note that our approach renders large 2.

- If IVGI≤64, then G is Hamiltonian.Taking up the theme regarding the number of Hamiltonian cycles in a Hamiltonian graph, Thomassen has provided a condition for a bipartite Hamiltonian graph to have a second Hamiltonian cycle.Theorem 5 (see). Let G be a Hamiltonian graph with bipartition A, B and let H be a Hamiltonian cycle in G
- Otherwise the graph is divided into connected components. Each component is a sole vertex, a cycle or a simple path. If there exists a cycle, then it must pass through all vertices, otherwise we won't be able to complete the Hamiltonian cycle. If this is the case, the answer is 2. (The cycle can be traversed in 2 directions.) Otherwise the answer is 0
- If bipartite graph has a Hamiltonian cycle, then is balanced. Let be a complete bipartite graph with vertex bipartition and ; then has Hamiltonian cycle if and only if ; that is, is balanced. Let denote a balanced complete bipartite graph. Corollary 8. Every Hamiltonian orientation of balanced complete bipartite graph has a dicycle cover with . This bound is best possible
- 2-factors with k cycles in Hamiltonian graphs Matija Bucić a, Erik Jahn , Alexey Pokrovskiyb, Benny Sudakova,1 a Department of Mathematics, ETH Zurich, Switzerland b Department of Economics, Mathematics, and Statistics, Birkbeck, UK a r t i c l e i n f o a b s t r a c t Article history: Received 24 May 2019 Available online 10 March 202
- ing if a.
- g two parts G (L) and its copy G (R) with original graph edges replaced by corresponding L-> R edges
- imum degree at least n=2 contains a Hamilton cycle. This

It turns out that complete graphs and odd cycles are the only graphs with χ(G)=∆(G)+1. Theorem 3 (Brooks). χ(G)≤ ∆(G)unless G is the complete graph or an odd cycle. Applications of colouring: schedulling, wireless communication, job assignment, and many more... 3.1. Edge-colouring We can similarly colour edges of a graph Below is an example of the complete bipartite graph $K_{5, 3}$: Number of Vertices, Edges, and Degrees in Complete Bipartite Graphs Since there are $r$ vertices in set $A$ , and $s$ vertices in set $B$ , and since $V(G) = A \cup B$ , then the number of vertices in $V(G)$ is $\mid V(G) \mid = r + s$ Disjoint Hamiltonian Cycles in Bipartite Graphs Michael Ferrara1, Ronald Gould1, Gerard Tansey 1 Thor Whalen2 Abstract Let G = (X,Y) be a bipartite graph and deﬁne σ2 2 (G) = min{d(x) + d(y) : xy /∈ E(G), x ∈ X, y ∈ Y}. Moon and Moser [6] showed that if G is a bipartite graph on 2n vertices such that σ2 MAT 145: PROBLEM SET 4 3 (c)Show that the complete bipartite graph K n;m admits a Hamiltonian cycle if and only if n = m. Solution. (a)Fix a given vertex v 1, then since K n is the complete graph it is connected to the other (n 1) vertices We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings in dense bipartite graphs. As a consequence, a new bound on the number of Hamiltonian cycles in a directed graph is proved, by using the ratio of the number of 1-factors. Based on this bound, we prove that our algorithm runs in expected time of $O(n^{8.5.

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply fa For each n=3 the cycle graph C n is Hamiltonian 4. For each n=3, the complete graph K n is Hamiltonian 5. For each n ≥ 2 the complete bipartite graph K n,n is Hamiltonian. 4.5 Theorem: If in a simple connected graph with n vertices (where n ≥ 3) the degrees of every vertex is greater than or equal to n/2, then the graph is Hamiltonian. Proof: In a simple connected graph with n vertices. forms a Hamiltonian cycle of G. We show that the matching graph M(K n,n) of a complete bipartite graph is bipartite if and only if n is even or n = 1. We prove that M(K n,n) is connected for n even and M(K n,n) has two components for n odd, n ≥ 3. We also compute distances between perfect matchings in M(K n,n). 1 Introductio Answer to formula for number of hamiltonian cycles in a complete bipartite graph Kn,n with n>=2. please explain the solution.. Deogun, J.S. and G Steiner, Hamiltonian cycle is polynomial on cocomparability graphs, Discrete Applied Mathematics 39 ( 1992) 165-l 72. Finding a Hamiltonian path or a Hamiltonian cycle in a general graph are classic NP-complete prob- lems. In this paper we anncunce polynomial time solutions for these problems on cocomparabilit

* (e) Which cube graphs Q n have a Hamilton cycle? Solution*.For n = 2, Q 2 is the cycle C 4, so it is Hamiltonian. Assume that Q n 1 is Hamiltonian and consider the cube graph Q n. Let V 1 and V 2 be as deﬁned in part (c). The vertices of V 1 form the cube graph Q n 1 and so there is a cycle C covering all the vertices of V 1 Hamilton cycles in graphs and hypergraphs: an extremal perspective Abstract. As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to.

Complete Bipartite Graph Example- The following graph is an example of a complete bipartite graph- Here, This graph is a bipartite graph as well as a complete graph. Therefore, it is a complete bipartite graph. This graph is called as K 4,3. Bipartite Graph Chromatic Number- To properly color any bipartite graph, Minimum 2 colors are required * Alternating Hamiltonian cycles in two coloured complete bipartite graphs*. / Chetwynd, Amanda G.; Hilton, A. J. W. In: Journal of Graph Theory, Vol. 16, No. 2, 06.1992. Determine whether a given graph contains Hamiltonian Cycle or not. If it contains, then prints the path. Following are the input and output of the required function. Input: A 2D array graph[V][V] where V is the number of vertices in graph and graph[V][V] is adjacency matrix representation of the graph. A value graph[i][j] is 1 if there is a.

two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time. Keywords: bipartite permutation graphs; chordal bipartite graphs; combinatorial reconﬁguration; Hamiltonian cycle; PSPACE-complete; split graphs; strongly chordal graphs; unit interval. **A** **graph** is uniquely **hamiltonian** if it has exactly one **hamiltonian** **cycle**. According to a conjecture there are no $r$-regular uniquely **hamiltonian** **graphs** for $r > 2$ and of special interest is the. a Hamiltonian cycle is NP-complete, even if the height of the grid is restricted to 2 vertices. We give a polyno- mial time algorithm for deciding if a solid square grid graph admits a Hamiltonian cycle which visits vertices at most twice and turns at every vertex. 1 Introduction The Hamiltonian cycle problem (HCP) in grid graphs has been well studied and has led to application in nu-merous NP. The complete graph above has four vertices, so the number of Hamilton circuits is: (N - 1)! = (4 - 1)! = 3! = 3*2*1 = 6 Hamilton circuits. However, three of those Hamilton circuits are the same circuit going the opposite direction (the mirror image)

Finding a Hamiltonian cycle in a graph is one of the classical NP-complete problems. Complexity of the Hamiltonian problem in permutation graphs has been a well-known open problem. In this paper the authors settle the complexity of the Hamiltonian problem in the more general class of cocomparability graphs. It is shown that the Hamiltonian cycle existence problem for cocomparability graphs is i A cycle in a graph is a path that begins and ends at the same vertex. v 1v 2 ···v kv 1 Cycles are also called circuits. We deﬁne paths and cycles for directed graphs analogously. A path or cycle is called simple if no vertex is traversed more than once. From now on we will only consider simple paths and cycles. 17/56. Graphs CSE235 Introduction Types Classes Bipartite Graphs. * Graph of minimal distances*. Check to save. Show distance matrix. Distance matrix. Select a source of the maximum flow. Select a sink of the maximum flow. Maximum flow from %2 to %3 equals %1. Flow from %1 in %2 does not exist. Source. Sink. Graph has not Hamiltonian cycle. Graph has Hamiltonian cycle. Graph has not Hamiltonian path. Graph has.

* The complete bipartite graph consists of two partite sets and containing and elements respectively with all is bipartite*. 2. Every cycle of is even. 3. . Proof:: Suppose is bipartite and there is a cycle which has an odd number of edges. Now for odd is in the same partite set as . But since the cycle is odd and are in the same partite set which is a contradiction. Conversely suppose has no. Journal of Graph Theory: Issue number: 2: Volume: 16: Number of pages: 6: Pages (from-to) 153-158: Publication Status : Published <mark>Original language</mark> English: Abstract. We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors. Home; Study; Research; Collaborate; Global.

It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, Nishizeki, and Saito, J. Inform. Proc. 1980) or to test whether a Hamiltonian cycle exists in a 4-regular graph, even when it is the graph formed by an arrangement of Jordan curves (Iwamoto and. Spanning Cycles through Speci ed Edges in Bipartite Graphs Reza Zamani and Douglas B. Westy Abstract P osa proved that if Gis an n-vertex graph in which any two nonadjacent vertices have degree sum at least n+ k, then Ghas a spanning cycle containing any speci ed family of disjoint paths with a total of kedges. We consider the analogous problem for a bipartite graph Gwith nvertices and parts. Alternating hamiltonian cycles in two colored complete bipartite graphs Alternating hamiltonian cycles in two colored complete bipartite graphs Chetwynd, A. G.; Hilton, A. J. W. 1992-06-01 00:00:00 We give necessary and sufficient conditions for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edge set is colored with two colors

for every cubic planar bipartite graph nding a second Hamiltonian cycle can be found in time linear in the number of vertices via a standard pivoting algorithm. We fail to settle the conjecture, but we prove it for cubic planar bipartite WH(6)-minor free graphs. iii. Acknowledgements First and foremost, I thank my supervisor Laura Sanit a for her incredible guidance and mentorship. This work. (1989) also characterized Hamiltonian bipartite tournaments, but they used a result on Hamiltonian cycles in 2-edge-colored complete graphs by Bánkfalvi and Bánkfalvi (1968). In Contreras-Balbuena et al. (2017), we gave a sufﬁcient condition for a 2-edge-colored multigraph to possess an alternating Hamiltonian cycle. Let G be a 2-edge. However below graph contains even number of vertices (10 vertices), but still I cant guess the perfect matching - in which every vertex of of the graph is incident to exactly one edge. One of the matching can be $\{E_1,E_4,E_6\}$. However not every vertex (here, $\{V_5,V_6,V_9,V_{10}\}$) is incident to exactly one edge in this matching as required by above definition. So this is not definitely. Hence, it follows that the only possible 2-factor is a hamiltonian cycle. Further-more, this graph has n2/8+n/2+1/2 edges, demonstrating the extremal number of edges is at least this number. Now let G be a bipartite 2-factor hamiltonian graph of order n = 4m + 2 containing the extremal number of edges. Let C be a hamiltonian cycle in G wit

the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where 1Uk U jV-Gƒj 4. In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n.We show that a related degree condition. 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. 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 . As a result, instead of complete characterization. 1. Give an example of a graph that has a hamiltonian cycle but no eulerian circuit and vice versa. 2. The complete bipartite graph Km,n is a graph with m + n vertices. These vertices are divided into a set of size m and a set of size n. We call these sets the parts of the graph. Within each set, there are no edges, but between each pair of vertices in different sets, there is an edge (2) By deleting any one edge from Hamiltonian cycle,we can get Hamiltonian path. (3) A graph may contain more than one Hamiltonian cycle. (4) A complete graph kn, will always have a Hamiltonian cycle, when n>=3 . Note : We don't have simple necessary and sufficient criteria for the existence of Hamiltonian cycles. However, we have many theorems. Detection of Hamiltonian circuits in a directed graph - Volume 24 Issue 2. To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account

DOI: 10.1002/JGT.3190160206 Corpus ID: 40044813. Alternating hamiltonian cycles in two colored complete bipartite graphs @article{Chetwynd1992AlternatingHC, title={Alternating hamiltonian cycles in two colored complete bipartite graphs}, author={A. Chetwynd and A. Hilton}, journal={J. Graph Theory}, year={1992}, volume={16}, pages={153-158} graph when it is clear from the context) to mean an isomorphism class of graphs. Important graphs and graph classes De nition. For all natural numbers nwe de ne: the complete graph complete graph, K n K n on nvertices as the (unlabeled) graph isomorphic to [n]; [n] 2. Complete graphs correspond to cliques. for n 3, the cycle Of course, as with more general graphs, there are bipartite graphs with few edges and a Hamilton cycle: any even length cycle is an example. We note that, in general, a complete bipartite graph K m, n is a bipartite graph with | X | = m, | Y | = n, and every vertex of X is adjacent to every vertex of Y A hamilton cycle in a graphT is a spanning cycle ofT.IfS is a set of edge-disjoint hamilton cycles inT and if E(S) isthesetofedgesoccurringinthehamiltoncyclesinS,thenSissaidtobemaximalifT −E(S)hasnohamiltoncycle. In 1993, Hoffman et al. [5] showed that there exists a maximal set S of m edge-disjoint hamilton cycles in Kn i Partition of Hamiltonian Biagraph into independent cycles In this sequel we will discuss finite simple graph, (Bondy & Chvatal, 1976) illustrate that if G is a bipartite graph Kn,n with bipartion (A,B) and for any x ϵ A , yϵ B and d(x) + d(y) ≥ n+1 Then G is Hamiltonian graph Our main target is to prove the following result; Theorem 4.1

The cycle spectrum of a graph G is the set of lengths of cycles in G. A cycle containing all vertices of G is a spanning or Hamiltonian cycle, and a graph having such a cycle is a Hamiltonian graph. An n-vertex graph is pancyclic if its cycle spectrum is {3,...,n}. All our graphs have no loops or multiple edges 2-factors with k cycles in Hamiltonian graphs Matija Buci ć a, Erik Jahn , Alexey Pokrovskiyb, Benny Sudakova,1 a Department of Mathematics, ETH Zurich, Switzerland b Department of Economics, Mathematics, and Statistics, Birkbeck, UK a r t i c l e i n f o a b s t r a c t Article history: Received 24 May 2019 Available online 10 March 2020. Earlier, Saburo developed a field theoretic approximation of the number of Hamiltonian cycles in graphs C m C n in as well as in planar random lattices . Fireze et al. considered generating and counting Hamiltonian cycles in random regular graphs . Given these results, we note that our approach renders large 2-crossing-critical graphs to be the first nonplanar graph family for which the number of Hamiltonian cycles can be exactly determined. It may be relevant that the dissertatio Abstract: We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating

Proof: If is an even number then the graph is a bipartite graph [3]. Hence is a bipartite graph. (By using the above Result). Theorem 2.9: The graph is a bipartite graph if are distinct odd primes. Proof: Suppose is an odd number. Then is not a bipartite graph because it contains an odd cycle. Hence is a bipartite graph if is a bipartite graph It is easy to see that any bipartite graph is two colorable and vice-versa. Simply take the set and color it red and color the set green. Likewise, if the graph can be colored using two colors, define as the red colored nodes and as the green nodes. In particular, we say that the chromatic number of any bipartite graph is 2 A chain or circuit in a graph is said to be hamiltonian if each vertex of the graph appears in it precisely once. Paths and cycles of digraphs are called hamiltonian if the same condition holds. A graph containing a hamiltonian circuit, or a digraph containing a hamiltonian cycle is referred to as a hamiltonian graph or digraph 2. No tree is Hamiltonian 3. For each n=3 the cycle graph C n is Hamiltonian 4. For each n=3, the complete graph K n is Hamiltonian 5. For each n ≥ 2 the complete bipartite graph K n,n is Hamiltonian. 4.5 Theorem: If in a simple connected graph with n vertices (where n ≥ 3) the degrees of every vertex is greater than or equal to n/2, then the graph is Hamiltonian. Proof Hall's Marriage Theorem and **Hamiltonian** **Cycles** **in** **Graphs** Lionel Levine May, 2001 If S is a set of vertices in a **graph** G, let d(S) be the **number** **of** vertices in G adjacent to at least one member of S. The following result is known as Phillip Hall's marriage theorem. Proposition 1 (Hall, 1935). Suppose **A**, B, |**A**| = |B| = n are the parts of a **bipartite** **graph** with the property that |d(S)| ≥ |S.