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Finding Uniquely Hamiltonian Graphs of Minimum Degree Three with Small Crossing Numbers
[chapter]

2016
*
Lecture Notes in Computer Science
*

In this work we try to

doi:10.1007/978-3-319-39636-1_1
fatcat:idccsk4wubd5fbvuomkoslostu
*find**uniquely**hamiltonian**graphs**with**minimum**degree**three*and a*small**crossing**number*by minimizing the*number**of**crossings*in an embedding and the*number**of**degree*-two vertices ... Although our implementation could not*find*a*uniquely**hamiltonian*planar*graph**with**minimum**degree**three*disproving Bondy and Jackson's conjecture, we were able to*find**uniquely**hamiltonian**graphs**of**minimum*... Conclusions and Future Work In this paper we presented a new optimization problem for*finding**uniquely**hamiltonian**graphs**of**minimum**degree**three**with**small**crossing**numbers*. ...##
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Page 2901 of Mathematical Reviews Vol. , Issue 2001E
[page]

2001
*
Mathematical Reviews
*

Other special classes

*of**small*order*graphs*are shown to be*Hamiltonian*, such as 3-connected cubic planar*graphs**with*at most 176 vertices and*with*face size at most 6. ... The author proves that if d,({x, x’}) > n/2+2 for each pair x,x’ € X, and the*minimum**degree**of*the induced*graph*(X) is at least*three*, then X is covered by one cycle. ...##
###
Regular graphs with few longest cycles
[article]

2022
*
arXiv
*
pre-print

We prove that for every d ≥ 3 there is an infinite family

arXiv:2104.10020v2
fatcat:3lnped7sebch7avfwpmesvjbci
*of**hamiltonian*3-connected*graphs**with**minimum**degree*d,*with*a bounded*number**of**hamiltonian*cycles. ... It is shown that if a 3-regular*graph*G has a*unique*longest cycle C, at least two components*of*G - E(C) have an odd*number**of*vertices on C, and that there exist 3-regular*graphs**with*exactly two such ... The research presented in this paper was supported by a Postdoctoral Fellowship*of*the Research Foundation Flanders (FWO). ...##
###
Hamiltonian Connectedness of Recursive Dual-Net

2009
*
2009 Ninth IEEE International Conference on Computer and Information Technology
*

The dual-construction extends a symmetric

doi:10.1109/cit.2009.17
dblp:conf/IEEEcit/LiPC09
fatcat:bhxtzpgfyvd6jaxiolqclnfm3i
*graph*G into a symmetric*graph*G*with*size 2n 2 and node-*degree*d + 1, where n and d are the size and the node-*degree**of*G, respectively. ... Therefore, a k-level Recursive Dual-Net RDN k (B) contains (2n 0 ) 2 k /2 nodes*with*a node*degree**of*d 0 + k, where n 0 and d 0 are the size and the node-*degree**of*the base network B, respectively. ... That is, the*minimum**number**of*links those removal will disconnect two halves equals to half*of*the total*number**of**cross*-edges*of*level k which is (2n 0 ) 2 k /8 . ...##
###
Every 4-Connected Graph with Crossing Number 2 is Hamiltonian

2018
*
SIAM Journal on Discrete Mathematics
*

A seminal theorem

doi:10.1137/17m1138443
fatcat:wycdxm4lwfaxzkdnl7r47ayzaa
*of*Tutte states that planar 4-connected*graphs*are*Hamiltonian*. Applying a result*of*Thomas and Yu, one can show that every 4-connected*graph**with**crossing**number*1 is*Hamiltonian*. ... We also discuss the traceability and Hamiltonicity*of*3-connected*graphs**with**small**crossing**number*and few 3-cuts, and present applications*of*our results. ... We thank Roman Soták for showing us the "flip trick" described in the first paragraph*of*the proof*of*Claim 1, as well as Robert Barish, who constructed several interesting examples, one*of*which settles ...##
###
Constructing Orthogonal de Bruijn Sequences
[chapter]

2011
*
Lecture Notes in Computer Science
*

A (σ, k)-de Bruijn sequence is a

doi:10.1007/978-3-642-22300-6_50
fatcat:s6gn6s3qmvc4fjj34rt264jlfa
*minimum*length string on an alphabet set*of*size σ which contains all σ k k-mers exactly once. ... Based on this approach, we present a heuristic which proves capable*of*efficiently constructing optimal collections*of*mutuallyorthogonal sequences for*small*values*of*σ and k, which supports our conjecture ... By the Ghouila-Houri theorem [23] , a directed*graph*D*with*n vertices has a*Hamiltonian*cycle if both the in-*degree*and out-*degree**of*every vertex*of*D are at least n/2. ...##
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Plane Geometric Graph Augmentation: A Generic Perspective
[chapter]

2012
*
Thirty Essays on Geometric Graph Theory
*

We consider augmentation problems over plane geometric

doi:10.1007/978-1-4614-0110-0_17
fatcat:vpjhqxc6qffe3h2tjjwzle6stq
*graphs*, that is,*graphs*given*with*a*crossing*-free straight-line embedding in the plane. ... The geometric constraints on the possible new edges render some*of*the simplest augmentation problems intractable, and in many cases only extremal results are known. ... If the augmentation is feasible, then the goal is to*find*the*minimum**number**of*new edges required. ...##
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On dispersability of some circulant graphs
[article]

2022
*
arXiv
*
pre-print

The matching book thickness mbt(G)

arXiv:2109.10163v2
fatcat:lnxkshg45jgitbazx2qjk2kc3m
*of*a*graph*G is the least*number**of*pages in a book embedding such that each page is a matching; G is dispersable if mbt(G) = Δ(G), where Δ is the maximum*degree*. ... We show that most circulant*graphs*G*with*jump lengths not exceeding 3 are dispersable or nearly dispersable. ... This would reduce the*number**of*purple edges from five to*three*. As n ≡ 3 (mod 4) increases, the sparse purple page stays at*three*edges. Hence, sparseness is 3, one more than the*minimum*. ...##
###
Rooted Cycle Bases

2017
*
Journal of Graph Algorithms and Applications
*

We show that in an unweighted or positively weighted

doi:10.7155/jgaa.00434
fatcat:lcmhmzbmnjhcda24z6fhdmscky
*graph*, it is possible to*find*the*minimum*weight rooted cycle basis in polynomial time. ... A cycle basis in an undirected*graph*is a minimal set*of*simple cycles whose symmetric differences include all Eulerian subgraphs*of*the given*graph*. ... The work*of*the first author was supported by the National Science Foundation under Grant CCF-1228639 and by the Office*of*Naval Research under Grant No. N00014-08-1-1015. ...##
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Rooted Cycle Bases
[article]

2015
*
arXiv
*
pre-print

We show that in an unweighted or positively weighted

arXiv:1504.04931v1
fatcat:atjkpge2bzggloobievhm5t4za
*graph*, it is possible to*find*the*minimum*weight rooted cycle basis in polynomial time. ... A cycle basis in an undirected*graph*is a minimal set*of*simple cycles whose symmetric differences include all Eulerian subgraphs*of*the given*graph*. ... The work*of*the first author was supported by the National Science Foundation under Grant CCF-1228639 and by the Office*of*Naval Research under Grant No. N00014-08-1-1015. ...##
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Context-based Space Filling Curves

2000
*
Computer graphics forum (Print)
*

The resulting one-dimensional representation

doi:10.1111/1467-8659.00413
fatcat:3vwmjlt4ivbdrpgblu2yeaxwta
*of*the image has improved autocorrelation compared*with*universal scans such as the Peano-Hilbert space filling curve. ... We also discuss the potential*of*improved autocorrelation*of*context-based space filling curves for image and video lossless compression. Dafner, Cohen-Or and Matias / Context-Based SFC ... The objective then is to*find*a*Hamiltonian*path over the grid*graph*for which the total weight is*small*. ...##
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Open questions on Tantrix graphs

2017
*
Mathematical Gazette
*

TantrixTM tiles are black hexagons imprinted

doi:10.1017/mag.2017.8
fatcat:2lxqz3xyjbbu3pzjgfvbhu3ni4
*with**three*coloured paths [1] joining pairs*of*edges. There are*three*different kinds*of*path. ... However, the two tiles consisting*of**three*straight lines (meeting at the centre*of*the hexagon) are not part*of*the set, so actually there are only 14 different tiles. ... If not,*find*a maximal planar subgraph. (7) Can you colour the vertices*of*your*graph*so that no pair*of*adjacent vertices has the same colour? If so, what is the*minimum**number**of*colours needed? ...##
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Exploration of the relationship between topology and designability of conformations

2011
*
Journal of Chemical Physics
*

*Unique*conformation giving

*minimum*energy is identified for each sequence and the

*number*

*of*sequences folding to each conformation (designability) is obtained. ... Complementary interaction

*graphs*are then generated for each conformation and are described using a set

*of*

*graph*features. ... , (vii)

*number*

*of*nodes

*with*

*minimum*

*degree*(n_min_d), (viii)

*number*

*of*nodes

*with*maximum

*degree*(n_max_d), (ix)

*number*

*of*nodes

*with*average

*degree*(n_avg_d), (x)

*number*

*of*nodes

*with*

*minimum*shortest ...

##
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Intersection Graphs of Non-crossing Paths
[article]

2020
*
arXiv
*
pre-print

We study

arXiv:1907.00272v2
fatcat:paeipvyuk5gmxl6wcunfatj56a
*graph*classes modeled by families*of*non-*crossing*(NC) connected sets. Two classic*graph*classes in this context are disk*graphs*and proper interval*graphs*. ... Finally, each such*graph*G is shown to have a*Hamiltonian*cycle if and only if it is 2-connected, and when G is not 2-connected, a*minimum*-leaf spanning tree*of*G has ℓ leaves if and only if G's block-cutpoint ... In the*minimum*connected dominating set (MCDS) problem, the input is a*graph*G, and the goal is to*find*a connected dominating set*with*the fewest vertices. ...##
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Mapping NP-Hard Problems to Restricted Adiabatic Quantum Architectures
[article]

2019
*
arXiv
*
pre-print

For each

arXiv:1911.00249v2
fatcat:d4c4r3gaqbc7pk3jjlusngnz6u
*of*the problem mapping methods presented, we examine how the physical qubit count scales*with*problem size on architectures*of*varying connectivity. ... The theory underpinning this framework provides tools to aid in the manipulation*of*Ising*Hamiltonians*for the purposes*of*Ising energy minimisation, and could be used to assist in developing and optimising ...*of**degree*D 0 to a*graph**of*at most*degree**three*. ...
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