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A proof of the dense version of Lovász conjecture

2011
*
Electronic Notes in Discrete Mathematics
*

We prove that every sufficiently large dense

doi:10.1016/j.endm.2011.09.047
fatcat:mcxy7drphfav3mctn6y2yg2ika
*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. ...##
###
Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

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. ...

##
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Magic and supermagic dense bipartite graphs

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. ...

##
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HAMILTONICITY OF CAMOUFLAGE GRAPHS

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

doi:10.33564/ijeast.2021.v05i10.022
fatcat:yylc75ifrng67cpwixa5m4jfti
*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 . ...##
###
Criticality of Counterexamples to Toroidal Edge-Hamiltonicity

2015
*
Graphs and Combinatorics
*

edge is on a

doi:10.1007/s00373-015-1542-5
fatcat:w3yiizppzvgyvk7po3s2ewyx34
*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*. ...##
###
Cubic Cayley Graphs and Snarks
[chapter]

2014
*
Fields Institute Communications
*

Combining the theory of Cayley maps with the existence of certain kinds of independent sets of vertices

doi:10.1007/978-1-4939-0781-6_2
fatcat:3zeawbc6lbdzpfomiwkmrxygwa
*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. ...##
###
The degrees, number of edges, spectral radius and weakly Hamilton-connectedness of bipartite graphs
[article]

2018
*
arXiv
*
pre-print

Note that any

arXiv:1809.05265v1
fatcat:uiwggczetrhufdlkclkyyq75yq
*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 ...##
###
Spectral Conditions for a Graph to be Hamilton-Connected

2013
*
Applied Mechanics and Materials
*

Some spectral conditions for a

doi:10.4028/www.scientific.net/amm.336-338.2329
fatcat:4cz4hyrwfzgp3lg66fwogv52uy
*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*. ...##
###
Rainbow Hamilton cycles in random regular graphs
[article]

2005
*
arXiv
*
pre-print

A random d-

arXiv:math/0508145v1
fatcat:t3nzbhkcvrf5vne5beggsahcva
*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. ...##
###
Hamilton decompositions of line graphs of some bipartite graphs

2005
*
Discussiones Mathematicae Graph Theory
*

Some

doi:10.7151/dmgt.1283
fatcat:5dxjohm7nzdd7epcodfn6s3xvm
*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*. ...##
###
Hamilton cycles in tensor product of graphs

1998
*
Discrete Mathematics
*

The relationship between the bieulerian orientation of a 4-

doi:10.1016/s0012-365x(97)00215-x
fatcat:6hpogomxe5fu5dzs2hduyt5mci
*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*. ...##
###
Rainbow Hamilton cycles in random regular graphs

2006
*
Random structures & algorithms (Print)
*

A random d-

doi:10.1002/rsa.20146
fatcat:gavov7rdk5bktduyibwhjhtqiy
*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. ...##
###
A survey on Hamilton cycles in directed graphs
[article]

2010
*
arXiv
*
pre-print

We survey some recent results on long-standing conjectures regarding

arXiv:1006.0590v1
fatcat:vayfrvmx2vba7fnbrhoriu2dxq
*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. ...##
###
A polynomial-time algorithm to determine (almost) Hamiltonicity of dense regular graphs
[article]

2020
*
arXiv
*
pre-print

We give a polynomial-time algorithm for detecting very long

arXiv:2007.14502v1
fatcat:dgaoqjqvsnfghfj7flo4ypquvi
*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*. ...##
###
The robust component structure of dense regular graphs and applications

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*. ...

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