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A proof of the dense version of Lovász conjecture

Demetres Christofides, Jan Hladký, Andras Máthé
2011 Electronic Notes in Discrete Mathematics  
We prove that every sufficiently large dense connected vertex-transitive graph is Hamiltonian.  ...  The case where we are far from bipartiteness is significantly easier but even when we are close to bipartiteness we can still find the Hamilton cycle using the information provided in the above result.  ...  This enables us to split the proof into two parts according to whether the graph (or rather the iron-connected pieces of the graph we obtain from Lemma 2.1) is c 4 -close to bipartite or not.  ... 
doi:10.1016/j.endm.2011.09.047 fatcat:mcxy7drphfav3mctn6y2yg2ika

Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

Md. Khaliluzzaman, Md. Monirul Islam, Md. Monjur Hasan
2015 International Journal of Computer Applications  
In this paper, we study Hamiltonicity of 3-connected, 3-regular planar bipartite graph G with partite sets V=M  N. We shall prove that G has a Hamiltonian cycle if G is balanced with M = N.  ...  For that we present an algorithm for a bipartite graph K M,N where M>3, N>3 and M,N both are even to possess a Hamiltonian cycle.  ...  The existence of Hamilton cycles in 2-connected, k-regular graphs have been investigated in [5, 6] by various authors.  ... 
doi:10.5120/20596-3096 fatcat:tejwu5a2lva5toxq5s47sqxipi

Magic and supermagic dense bipartite graphs

Jaroslav Ivančo
2007 Discussiones Mathematicae Graph Theory  
In the paper we prove that any balanced bipartite graph with minimum degree greater than |V (G)|/4 ≥ 2 is magic. A similar result is presented for supermagic regular bipartite graphs.  ...  A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex  ...  In [4] there is proved that every 4-regular bipartite graph which can be decomposed into two edge-disjoint Hamilton cycles is supermagic. The following lemmas extend this result. Lemma 3.  ... 
doi:10.7151/dmgt.1384 fatcat:cxogtxxgwrdobk3vtoolcakf2a

HAMILTONICITY OF CAMOUFLAGE GRAPHS

Sowmiya K
2021 International Journal of Engineering Applied Sciences and Technology  
The well-known mathematician Barnette introduced the open conjecture which becomes a theorem by restricting our attention to the class of graphs which is 3-regular, 3- connected, bipartite, planar graphs  ...  This paper examines the Hamiltonicity of graphs having some hidden behaviours of some other graphs in it.  ...  THEOREM 2 Every connected vertex-transitive simple graph has a Hamilton path. Proof: Let G be a simple vertex transitive connected graph with .  ... 
doi:10.33564/ijeast.2021.v05i10.022 fatcat:yylc75ifrng67cpwixa5m4jfti

Criticality of Counterexamples to Toroidal Edge-Hamiltonicity

M. N. Ellingham, Emily A. Marshall
2015 Graphs and Combinatorics  
edge is on a hamilton cycle.  ...  One possible way to modify it is by characterizing the situations where some edge is not on a hamilton cycle.  ...  The first strengthening is to look for what are known as Tutte cycles instead of hamilton cycles, in 2-connected graphs instead of 4-connected graphs.  ... 
doi:10.1007/s00373-015-1542-5 fatcat:w3yiizppzvgyvk7po3s2ewyx34

Cubic Cayley Graphs and Snarks [chapter]

Ademir Hujdurović, Klavdija Kutnar, Dragan Marušič
2014 Fields Institute Communications  
Combining the theory of Cayley maps with the existence of certain kinds of independent sets of vertices in arc-transitive graphs, some new partial results are obtained suggesting promising future research  ...  directions in regards to this conjecture.  ...  The second author was supported in part by ARRS, P1-0285, J1-2055 and Z1-4006, and by ESF EuroGiga GReGAS.  ... 
doi:10.1007/978-1-4939-0781-6_2 fatcat:3zeawbc6lbdzpfomiwkmrxygwa

The degrees, number of edges, spectral radius and weakly Hamilton-connectedness of bipartite graphs [article]

Jia Wei, Zhifu You
2018 arXiv   pre-print
Note that any bipartite graph is not Hamilton-connected. We consider the weak version of Hamilton-connected property among bipartite graphs.  ...  In this paper, we present some degrees, number of edges, and spectral radius conditions for a simple balanced bipartite graph to be weakly Hamilton-connected.  ...  Spectral radius and weakly Hamilton-connected bipartite graphs In Section 4, we give some sufficient conditions for weakly Hamilton-connectedness of bipartite graphs in terms of spectral radius and signless  ... 
arXiv:1809.05265v1 fatcat:uiwggczetrhufdlkclkyyq75yq

Spectral Conditions for a Graph to be Hamilton-Connected

Gui Dong Yu, Yi Zheng Fan
2013 Applied Mechanics and Materials  
Some spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or signless Laplacian of the graph or its complement are established, and then the  ...  condition on the signless Laplacian spectral radius of a graph for the existence of Hamiltonian paths or cycles is given.  ...  Lemma 2.6 [5] Let G be a connected graph. Then, γ(G) ≤ max{d(v) + m(v) : v ∈ V (G)}, (2. 3) with equality if and only if G is a regular graph or a bipartite semiregular graph.  ... 
doi:10.4028/www.scientific.net/amm.336-338.2329 fatcat:4cz4hyrwfzgp3lg66fwogv52uy

Rainbow Hamilton cycles in random regular graphs [article]

Svante Janson, Nicholas Wormald
2005 arXiv   pre-print
A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is  ...  A rainbow subgraph of an edge-coloured graph has all edges of distinct colours.  ...  In particular, rainbow Hamilton cycles in the multigraph correspond to Hamilton cycles in the bipartite graph that obey this traffic rule.  ... 
arXiv:math/0508145v1 fatcat:t3nzbhkcvrf5vne5beggsahcva

Hamilton decompositions of line graphs of some bipartite graphs

Dawid A. Pike
2005 Discussiones Mathematicae Graph Theory  
Some bipartite Hamilton decomposable graphs that are regular of degree δ ≡ 2 (mod 4) are shown to have Hamilton decomposable line graphs.  ...  One consequence is that every bipartite Hamilton decomposable graph G with connectivity κ(G) = 2 has a Hamilton decomposable line graph L(G).  ...  Let G be a bipartite graph that is regular of degree δ = 4k+2.  ... 
doi:10.7151/dmgt.1283 fatcat:5dxjohm7nzdd7epcodfn6s3xvm

Hamilton cycles in tensor product of graphs

R. Balakrishnan, P. Paulraja
1998 Discrete Mathematics  
The relationship between the bieulerian orientation of a 4-regular graph G and the existence of a pair of edge-disjoint Hamilton cycles in G®K2 is established.  ...  In this paper, we characterize graphs G for which G®K2 is Hamiltonian, where ® denotes the tensor product of graphs.  ...  But the graph of Fig. 3 is not only 2-connected but also has a pair of edge-disjoint Hamilton cycles.  ... 
doi:10.1016/s0012-365x(97)00215-x fatcat:6hpogomxe5fu5dzs2hduyt5mci

Rainbow Hamilton cycles in random regular graphs

Svante Janson, Nicholas Wormald
2006 Random structures & algorithms (Print)  
A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8.  ...  A rainbow subgraph of an edge-coloured graph has all edges of distinct colours.  ...  In particular, rainbow Hamilton cycles in the multigraph correspond to Hamilton cycles in the bipartite graph that obey this traffic rule.  ... 
doi:10.1002/rsa.20146 fatcat:gavov7rdk5bktduyibwhjhtqiy

A survey on Hamilton cycles in directed graphs [article]

Daniela Kühn, Deryk Osthus
2010 arXiv   pre-print
We survey some recent results on long-standing conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments.  ...  by a set of Hamilton cycles which are 'almost' edge-disjoint.  ...  A (3s − 1)-regular 2-connected graph G on n = 9s + 2 vertices with no Hamilton cycle. To construct G, start with 3 disjoint cliques on 3s vertices each.  ... 
arXiv:1006.0590v1 fatcat:vayfrvmx2vba7fnbrhoriu2dxq

A polynomial-time algorithm to determine (almost) Hamiltonicity of dense regular graphs [article]

Viresh Patel, Fabian Stroh
2020 arXiv   pre-print
We give a polynomial-time algorithm for detecting very long cycles in dense regular graphs.  ...  whether G contains a cycle on at least n - c vertices.  ...  A Hamilton cycle in a graph is a spanning cycle, i.e. a cycle that contains every vertex of a graph.  ... 
arXiv:2007.14502v1 fatcat:dgaoqjqvsnfghfj7flo4ypquvi

The robust component structure of dense regular graphs and applications

Daniela Kühn, Allan Lo, Deryk Osthus, Katherine Staden
2014 Proceedings of the London Mathematical Society  
In this paper, we study the large-scale structure of dense regular graphs.  ...  (ii) We prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity. The 2-connected case of this was conjectured by Bondy and proved by Wei.  ...  ∈ N with t ≥ 2, there are infinitely many D ∈ N such that there exists a bipartite graph on 8D + 2 vertices which is D-regular and t-connected but does not contain a Hamilton cycle.  ... 
doi:10.1112/plms/pdu039 fatcat:mzwruo6eczhptlto5siwdcw74u
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