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

Benedikt Klocker, Herbert Fleischner, Günther R. Raidl
2016 Lecture Notes in Computer Science  
In this work we try to 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.  ... 
doi:10.1007/978-3-319-39636-1_1 fatcat:idccsk4wubd5fbvuomkoslostu

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]

Carol T. Zamfirescu
2022 arXiv   pre-print
We prove that for every d ≥ 3 there is an infinite family 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).  ... 
arXiv:2104.10020v2 fatcat:3lnped7sebch7avfwpmesvjbci

Hamiltonian Connectedness of Recursive Dual-Net

Yamin Li, Shietung Peng, Wanming Chu
2009 2009 Ninth IEEE International Conference on Computer and Information Technology  
The dual-construction extends a symmetric 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 .  ... 
doi:10.1109/cit.2009.17 dblp:conf/IEEEcit/LiPC09 fatcat:bhxtzpgfyvd6jaxiolqclnfm3i

Every 4-Connected Graph with Crossing Number 2 is Hamiltonian

Kenta Ozeki, Carol T. Zamfirescu
2018 SIAM Journal on Discrete Mathematics  
A seminal theorem 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  ... 
doi:10.1137/17m1138443 fatcat:wycdxm4lwfaxzkdnl7r47ayzaa

Constructing Orthogonal de Bruijn Sequences [chapter]

Yaw-Ling Lin, Charles Ward, Bharat Jain, Steven Skiena
2011 Lecture Notes in Computer Science  
A (σ, k)-de Bruijn sequence is a 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.  ... 
doi:10.1007/978-3-642-22300-6_50 fatcat:s6gn6s3qmvc4fjj34rt264jlfa

Plane Geometric Graph Augmentation: A Generic Perspective [chapter]

Ferran Hurtado, Csaba D. Tóth
2012 Thirty Essays on Geometric Graph Theory  
We consider augmentation problems over plane geometric 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.  ... 
doi:10.1007/978-1-4614-0110-0_17 fatcat:vpjhqxc6qffe3h2tjjwzle6stq

On dispersability of some circulant graphs [article]

Samuel S. Joslin, Paul C. Kainen, Shannon Overbay
2022 arXiv   pre-print
The matching book thickness mbt(G) 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.  ... 
arXiv:2109.10163v2 fatcat:lnxkshg45jgitbazx2qjk2kc3m

Rooted Cycle Bases

David Eppstein, J. Michael McCarthy, Brian E. Parrish
2017 Journal of Graph Algorithms and Applications  
We show that in an unweighted or positively weighted 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.  ... 
doi:10.7155/jgaa.00434 fatcat:lcmhmzbmnjhcda24z6fhdmscky

Rooted Cycle Bases [article]

David Eppstein, J. Michael McCarthy, Brian E. Parrish
2015 arXiv   pre-print
We show that in an unweighted or positively weighted 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.  ... 
arXiv:1504.04931v1 fatcat:atjkpge2bzggloobievhm5t4za

Context-based Space Filling Curves

Revital Dafner, Daniel Cohen-Or, Yossi Matias
2000 Computer graphics forum (Print)  
The resulting one-dimensional representation 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.  ... 
doi:10.1111/1467-8659.00413 fatcat:3vwmjlt4ivbdrpgblu2yeaxwta

Open questions on Tantrix graphs

Heidi Burgiel, Mahmoud El-Hashash
2017 Mathematical Gazette  
TantrixTM tiles are black hexagons imprinted 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?  ... 
doi:10.1017/mag.2017.8 fatcat:2lxqz3xyjbbu3pzjgfvbhu3ni4

Exploration of the relationship between topology and designability of conformations

Sumudu P. Leelananda, Fadi Towfic, Robert L. Jernigan, Andrzej Kloczkowski
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  ... 
doi:10.1063/1.3596947 pmid:21702580 pmcid:PMC3133807 fatcat:odkmqseqnvfr3kqudtrpjyoblu

Intersection Graphs of Non-crossing Paths [article]

Steven Chaplick
2020 arXiv   pre-print
We study 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.  ... 
arXiv:1907.00272v2 fatcat:paeipvyuk5gmxl6wcunfatj56a

Mapping NP-Hard Problems to Restricted Adiabatic Quantum Architectures [article]

Gary J. Mooney, Sam U.Y. Tonetto, Charles D. Hill, Lloyd C.L. Hollenberg
2019 arXiv   pre-print
For each 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.  ... 
arXiv:1911.00249v2 fatcat:d4c4r3gaqbc7pk3jjlusngnz6u
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