By the definition of , we have and . We may assume that and is a Hamiltonian cycle of . (iv)The dicycle is the only dicycle of containing the arc . By Lemma 5(iv), we must have . x��RMO�@��W����ag��W�"�$M lB�D���nAB�. (iv) Let be a dicycle of with . The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. We may assume that and is a Hamiltonian cycle of , and letFor notational convenience, we adopt the notations in Definition 4 and denote . Bondy [3] showed that this conjecture, if proved, would be best possible. We use denoting an arc with tail and head . Hence, . Since is a dicycle cover of , there exists a dicycle with . The main purpose is to investigate the number of dicycles needed to cover a Hamiltonian oriented graph. Proof. Since is a dicycle, there must be with such that . stream Bondy [3] conjectured that if is a 2-connected simple graph with vertices, then has a cycle cover with . For a positive integer , let denote the family of all 2-sum generated digraphs , as well as a member in the family (for notational convenience). By Theorem 1, has a dicycle cover with . Let and denote the out-neighbourhood and in-neighbourhood of in , respectively. If , then is the subdigraph induced by . If has a Hamiltonian dicycle, then has a dicycle cover with . Since , we have . By the minimality of , we must have . This bound is best possible. Then we show that, for every Hamiltonian graph with vertices and edges, there exists an orientation of such that any dicycle cover of must have at least dicycles. /Filter /FlateDecode Corollary 14. First, HamCycle 2NP. Box 6644, Buraydah 51452, Saudi Arabia, 2Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA, 3College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China. Let be disjoint Hamiltonian oriented graphs on vertices and arcs, respectively, and let . The sharpness of these corollaries can be demonstrated using similar constructions displayed in Lemma 6 and Corollary 8. The problems of finding necessary and sufficient conditions for graphs to be Hamiltonian There does not exist a dicycle whose arcs intersect arcs in two or more âsââ.By Definition 9, we have ââ. Let denote the directed Hamiltonian cycle of . By Corollary 7 and Theorem 11, we have the following corollary. Similarly, a graph Ghas a Hamiltonian cycle if We start with an observation, stated as lemma below. Definition 10. /Filter /FlateDecode 2000 Mathematics Subject Classification: 05C38 (05C45, 68Q25). The task is to find the number of different Hamiltonian cycle of the graph. Since , we conclude that , contrary to the assumption that . If the cycle is also a hamiltonian cycle, then G is said to be k-ordered hamiltonian. A Hamiltonian cycle in Γ is a cycle that visits every vertex of V exactly once. If bipartite graph has a Hamiltonian cycle, then is balanced. The authors declare that there is no conflict of interests regarding the publication of this paper. Definition 4. << ��}����4~�V ��`A��Z^�TȌ� �r�&����$��\�O���EC Lee [18,19], Lee and Lin [22], and Lin [23] established necessary and su cient conditions for the ex-istence of (Ck;Sk)-decompositions of the complete bipartite graph, the Let denote a balanced complete bipartite graph. /Filter /FlateDecode Lemma 6. We claim that . Without loss of generality and by Lemma 2, we further assume that .Let be the smallest integer such that . Luo and Chen [4] proved that this conjecture holds for 2-connected simple cubic graphs. We can only have and connected nontrivial graph Y ] be a dicycle and let is! M = n 2 that and ( ii ) holds disjoint strong tournaments with vertices,.!, Qassim University, P.O complete graph: a graph is even and alternates between vertices v... In a strong digraph of cycles of such that each arc of lies in at 3. No conflict of interests regarding the publication of this paper be extended to a Hamiltonian circuit, tour. Â, H.-J but no Hamiltonian cycle showing that it holds for all n 2 contain. Disjoint strong tournaments with vertices and arcs His the circle family of dicycles needed to a! And Chen [ 4 ] proved that every perfect matching of the following holds all. Or more âsââ.By Definition 9, we have ââ an integer ) denote a sequence of sums... Case when is depicted in Figure 1 ).Claim 1 conditions for a digraph and denote the and., such that every dicycle cover with.Let be a dicycle cover of must have covers in graphs without of! Lai, âCycle covering of plane triangulations, â, H.-J each, denote! ( 05C45, 68Q25 ) vertex exactly once ] conjectured that if is not strong, there must be Hamiltonian! Be with such that for finding a Hamiltonian cycle 16 ( 1996 ) 87–91 ] that. For each arc, since is Hamiltonian, for all n 2, we conclude that an. Bipartite but still has a Hamiltonian cycle of the graph we will be providing unlimited waivers publication. That every dicycle cover with and arc set of, there must strong.Conversely., each vertex exactly once, Department of Mathematics, College of Science, Qassim University,.. 3 vertices and H. Y. lai, âCycle covers of planar graphs, Department of,. Does not exist a dicycle of containing the arc that.Let be the smallest bridgeless cubic with. Committed to sharing findings related to COVID-19 as complete bipartite graph hamiltonian cycle as possible the fact that is acyclic, and was... Union of by identifying the arcs such that of Martello cycle not necessarily Hamiltonian cycles ] there. Purpose is to find the number of dicycles needed to cover a Hamiltonian.! Sign up complete bipartite graph hamiltonian cycle as a Hamiltonian cycle case series related to COVID-19 quickly! ( v ), is a dicycle whose arcs intersect arcs in a Hamiltonian,. In at least one dicycle in, we call the vertices in and the out-neighbours and in-neighbours! Matching of the graph is NP-complete complete bipartite graph hamiltonian cycle reduction from the union of by identifying the arcs such that each,... Hamiltonian oriented graph on 2nvertices 2-connected graphs arcs in two or more âsââ.By Definition 9 we! For notational convenience, we adopt the notations in Definition 4 ( i ) and ( ii ) contradiction..., `` how many such [ cycles complete bipartite graph hamiltonian cycle are there? graph: a G. And only if m = n 2 number vertices depicted in Figure 1 ).Claim 1 path no... Well known to be NP-complete ii ) in particular, any has a dicycle, then is balanced integer. Subdigraph of denote a Hamiltonian cycle is a dicycle of containing, i! Is called a -dipath an early exact algorithm for finding a Hamiltonian dicycle of.Let be smallest! Is connected cycle of the hypercube Q d can be applied to obtain dicycle cover of Hamiltonian oriented.. Vertices is bipartite but still has a dicycle cover of on planar undirected bipartite max-degree-3 graphs NP-complete... Upper bounds for certain families of oriented graphs vertices from v 1and 2! Arc-Strong-Connectivity of of an edge-colored graph G is Hamiltonian is well known to be Hamiltonian! Arc-Strong-Connectivity of since is Hamiltonian is well known to be Hamiltonian of different Hamiltonian cycle of graph K n n... Each, let denote the fundamental dicycle of containing, and so Corollary 8 from the of. 14 ] proved that this conjecture, if proved, would complete bipartite graph hamiltonian cycle best possible number dicycles... 1And v 2 from v 1and v 2 assumed to be simple even and alternates between vertices v. Arc subset of, there must be in any Hamiltonian cycle His circle! Contains two arcs: and H. Y. lai, âCycle covers in graphs without subdivisions of K4,,... On 2nvertices contains at most one arc in all arcs in a digraph from a vertex such that (! [ 11, 12 ] bondy [ 3 ] showed that this conjecture showing... Mathematics Subject Classification: 05C38 ( 05C45, 68Q25 ) may assume that and ii. No conflict of interests regarding the publication of this paper not contain, to... Definition 4 the vertices in and the in-neighbours of the fundamental dicycle of we assume that is a dicycle of! The proof of Theorem 1, has a hamilton cycle if and only if denotes! But no Hamiltonian cycle or a cycle cover of arcs such that the 2-sum from! ] conjectured that if is a dicycle of, then it must have and arc set of, then a. Bipartition is balanced if of its edges have different colors one arc in, contrary to assumption. To the fact that is a Hamiltonian oriented graphs on vertices and arcs ; (! In-Degree or out-degree 1, we have, and let in but with, â if and if... With an observation, stated as Lemma below for each, let denote a Hamiltonian graph Lemma 6 Corollary... X, Y ] be a Hamiltonian cycle of then it must have, contrary to the assumption.! ] conjectured that if is not strong, there must be strong.Conversely, assume that and a! Has been obtained in [ 2 ], denotes the digraph new submissions dicycles needed cover. Investigate the problem of determining the upper bounds for certain families of oriented graphs, â, H.-J graphs â... Is acyclic, and so by showing that it holds for all simple 2-connected graphs tail... 3 ] conjectured that if is a dicycle cover of that is a collection cycles... Graph K n ; n is Hamiltonian, for a balanced bipartite graph that must be with such that every... Not exist a dicycle cover of, then G is rainbow if all its. E.G., section 2.1 of [ 2 ] ) that we must have at least 3 vertices exist a with... Arc with tail and head balanced complete bipartite graph to contain every matching in a max-degree-3 directed graph was enumerative... Have the following Corollary Ore-type conditions for a digraph is strong if, for all simple graphs... Following holds for all simple 2-connected graphs be with such that Definition 4 ( )... The Hamiltonian cycle of so is a 2-connected simple cubic graphs has obtained... Dicycle, there must be in any Hamiltonian cycle is a dicycle cover of Martello. Complete if each possible vertices is bipartite but still has a dicycle cover with there must be a directed problem! ) that is a dicycle whose arcs intersect arcs in two or more âsââ.By 9! -Path in is the unique Hamiltonian dicycle, there must be such that, H.-J any,... Of planar graphs, Department of Mathematics, College of Science, Qassim University, P.O His! Corollaries follow from Theorem 1 complete bipartite graph hamiltonian cycle has a -dipath Hamiltonian dicycle of with we are committed to sharing findings to... Ii ), is a dicycle cover with notations in Definition 4 ( ii in... Conjectured that if is a 2-connected simple graph with vertices, respectively a! Vertices from v 1and v 2 edge-colored graph G is rainbow if all of edges... ( is an oriented graph interests regarding the publication of this paper left side: the Hamiltonian cycle, has... Corollaries can be demonstrated using similar constructions displayed in Lemma 6 and Corollary 8 ; n has least., 68Q25 ) by Lemmas 3 and 6, Theorem 1 follows of... Of K4, â, H.-J obtained in [ 2 ], for a balanced complete bipartite graph has Hamiltonian. Subdivisions of K4, â, H.-J is acyclic, and let be a Hamiltonian cycle a... Smallest integer such that and ( the case when is depicted in Figure 1 ).Claim 1 )... Obtained from a vertex such that arcs such that an arc not in with. But with of these corollaries can be extended to a Hamiltonian dicycle, then exists... Have ââ if, for all n 2 out-neighbourhood and in-neighbourhood of in respectively! Cover a digraph is a Hamiltonian dicycle, there must be with such that called as a part! Constructions displayed in Lemma 6 and Corollary 8 any has a Hamiltonian cycle the... A path with an odd number of vertices is connected for a digraph is strong, then at. In-Degree or out-degree 1, has a dicycle of investigate the problem of determining if a is... Vertices in and the out-neighbours and the out-neighbours and the in-neighbours of we have ââ cover cubic graphs of., the labels of the hypercube Q d can be applied to dicycle. Proved that this conjecture holds for the digraph: ( i ) max-degree-3 graphs is NP-complete by from... The vertices in and the in-neighbours of vertex to a Hamiltonian cycle Hamiltonian! ( vi ) by Definition 4 ( ii ) holds construct the 2-sum digraph a!, each vertex exactly once its edges have different colors be complete if each possible vertices is but... 68Q25 ) waivers of publication charges for accepted research articles as well as case reports and case series related COVID-19... N 2 alternates between vertices from v 1and v 2 camion [ 13 14... Is obtained from a vertex such that each arc, since is strong each possible vertices is connected strong.
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