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Covering a graph with cycles passing through given edges

1997
*
Journal of Graph Theory
*

We propose

doi:10.1002/(sici)1097-0118(199710)26:2<105::aid-jgt5>3.0.co;2-w
fatcat:ny6y3edqajclpl3naxa4ldokpq
*a*conjecture: for each integer k 2:: 2, there exists N (k) such that if G = (Vb \12; E) is*a*bipartite*graph**with*IV 1 1 = 1\121 = n 2: N(k) and d( x) + d(y) 2: n + k for each pair of non-adjacent ... ,ek be k independent*edges*in G such that ei is incident*with*Xi for all i E {I, . ... paths of G, G has*a*hamiltonian*cycle**passing**through*all the*edges*of F. ...##
###
A New Property of Hamilton Graphs
[article]

2015
*
arXiv
*
pre-print

Hamilton

arXiv:1508.00068v1
fatcat:pevd37rb5vcp5jrw7a6mbch5u4
*graphs*as being*a*necessary and sufficient condition characterized in the connectivity of the subgraph that induced from the*cycle*structure of*a**given**graph*. ...*A*Hamilton*cycle*is*a**cycle*containing every vertex of*a**graph*.*A**graph*is called Hamiltonian if it contains*a*Hamilton*cycle*. ... Every two spanning*cycle*-sets in the*given**graph*have the same Hamiltonian property*with*the*given**graph*. Proof.*Graph*G is*a*(N 3 , N 2 )-free*graph*. ...##
###
A safe and complete algorithm for metagenomic assembly

2018
*
Algorithms for Molecular Biology
*

This problem can be naturally formulated as finding

doi:10.1186/s13015-018-0122-7
pmid:29445416
pmcid:PMC5802251
fatcat:yciged7yijghvga5jnd7ha7jbm
*a*collection of circular walks of*a*directed*graph*G that together*cover*all nodes, or*edges*, of G. ... In the node-*covering*case, our algorithm runs in time O(m 2 + n 3 ), and in the*edge*-*covering*case it runs in time O(m 2 n); n and m denote the number of nodes and*edges*, respectively, of G. ... The set of*cycles*of S(G)*passing**through*z xy is*a*subset of the set of*cycles*of S(G)*passing**through*x. Therefore, all*cycles*of S(G)*passing**through*z xy contain S(w) as sub-walk. ...##
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Covering planar graphs with forests, one having bounded maximum degree

2009
*
Journal of combinatorial theory. Series B (Print)
*

Yu,

doi:10.1016/j.jctb.2008.07.004
fatcat:2fmtmjp6jvfezjifvai4yoym7e
*Covering*planar*graphs**with*forests, J. Combin. Theory Ser. B. 94 (2005) 147-158 ]. ... We prove that every planar*graph*has an*edge*partition into three forests, one having maximum degree at most 4. This answers*a*conjecture of Balogh, Kochol, Pluhár and Yu [J. Balogh, M. Kochol,*A*. ... If F e 3 had*a**cycle*, this*cycle*should either*pass**through*ux or not. In the first case, this would imply that there is*a**cycle*(*passing**through*u ) in F m 3 ⊂ T m . ...##
###
A97-approximation algorithm for Graphic TSP in cubic bipartite graphs

2016
*
Discrete Applied Mathematics
*

Our approach involves finding

doi:10.1016/j.dam.2015.10.038
fatcat:kydydtum6nbbhljrxzolvhpbwa
*a**cycle**cover**with*relatively few*cycles*, which we are able to do by leveraging the fact that all*cycles*in bipartite*graphs*are of even length along*with*our knowledge of ...*Given**a*cubic bipartite connected*graph*G*with*n vertices, there is*a*polynomial time algorithm that computes*a*spanning Eulerian multigraph H in G*with*at most 9 7 n*edges*. Corollary 1.2. ...*A*4/3-approximation for TSP on cubic 3-edgeconnected*graphs*. arXiv:1101.5586, 2011. ...##
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MacLane's theorem for arbitrary surfaces

2009
*
Journal of combinatorial theory. Series B (Print)
*

*Given*

*a*closed surface S, we characterise the

*graphs*embeddable in S by an algebraic condition asserting the existence of

*a*sparse generating set for their

*cycle*space. ... And we thank

*a*thoughtful referee for

*a*suggesting an improvement to our notion of sparseness at vertices. ... Therefore, as any family F of

*passes*of W

*through*v

*with*p∈F c(p) = ∅ induces

*a*

*cycle*in H (v), this can happen only when F

*covers*E(v). ...

##
###
Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles
[article]

2021
*
arXiv
*
pre-print

*through*any

*given*fixed number of vertices or

*edges*. ... We also recognize the disjoint paths problem as

*a*special case of finding disjoint

*cycles*, and present (randomized) parallel algorithms for finding

*a*shortest

*cycle*and shortest 2-disjoint

*cycles*

*passing*...

*Given*

*a*

*graph*G on n vertices and k marked

*edges*e 1 , . . . , e k , let F(e 1 , . . . , e k ) be the family of all l disjoint

*cycles*

*passing*

*through*e 1 , . . . , e k . ...

##
###
Cubic graphs with large circumference deficit
[article]

2013
*
arXiv
*
pre-print

Markström, On stable

arXiv:1310.1042v2
fatcat:trxtlyy67fhdxlvpw4tsjt2xve
*cycles*and*cycle*double*covers*of*graphs**with*large circumference, Disc. Math. 312 (2012), 2540--2544]. ... The circumference c(G) of*a**graph*G is the length of*a*longest*cycle*. ... We say that*a**cycle**passes**through*B if*a*portion of this*cycle*(*a*path)*passes**through*B. The resulting*graph*B has two properties interesting to us. ...##
###
Covers of Non-Almost-Finite Type Sofic Systems

1988
*
Proceedings of the American Mathematical Society
*

To do this in

doi:10.2307/2047495
fatcat:ulq7ngbxojburkkur4uzzco6uy
*a*general setting requires some technical groundwork, found in §3. The construction itself is*given*in §4. ... The authors asked whether*a*non-AFT sofic system has*a*finite collection C of*covers*such that every*cover*is intercepted by*a**cover*in C. ... Suppose to the contrary that two*covers*rm, Tn*with*to > n are intercepted by*a**cover*(E,7r): We may assume -k is*given*by an*edge*labeling of*a*directed*graph*G(E). ...##
###
Covers of non-almost-finite type sofic systems

1988
*
Proceedings of the American Mathematical Society
*

To do this in

doi:10.1090/s0002-9939-1988-0958076-3
fatcat:cp5q7h2h6jdqnlw5eynfx4vvme
*a*general setting requires some technical groundwork, found in §3. The construction itself is*given*in §4. ... The authors asked whether*a*non-AFT sofic system has*a*finite collection C of*covers*such that every*cover*is intercepted by*a**cover*in C. ... Suppose to the contrary that two*covers*rm, Tn*with*to > n are intercepted by*a**cover*(E,7r): We may assume -k is*given*by an*edge*labeling of*a*directed*graph*G(E). ...##
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On Complexity of the Optimal Recombination for the Travelling Salesman Problem
[chapter]

2011
*
Lecture Notes in Computer Science
*

There may be some room for improvement of the algorithms, proposed in [7] for the TSP on

doi:10.1007/978-3-642-20364-0_19
fatcat:3mvkkwlodrbzpfjpl6xivuxxwy
*graphs**with*vertex degrees at most 3 or 4 and forced*edges*, in terms of the running time. ... It is required to find*a*shortest Hamiltonian*cycle*(circulation) in the*given**graph*(digraph) that*passes*all forced*edges*(arcs). ... are connected to the TSP on*graphs*(digraphs)*with*bounded vertex degree, arbitrary positive*edge*(arc) weights and*a**given*set of forced*edges*(arcs). ...##
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A 9/7-Approximation Algorithm for Graphic TSP in Cubic Bipartite Graphs
[article]

2014
*
arXiv
*
pre-print

Our approach involves finding

arXiv:1311.3640v2
fatcat:fhfji6h7bjhxxo4pwttgmiatoe
*a**cycle**cover**with*relatively few*cycles*, which we are able to do by leveraging the fact that all*cycles*in bipartite*graphs*are of even length along*with*our knowledge of ... Specifically, we provide*a*polynomial-time 9/7-approximation algorithm for cubic bipartite*graphs*and*a*(9/7+1/21(k-2))-approximation algorithm for k-regular bipartite*graphs*, both of which are improved ... One*cycle**passes**through*gadget If*a*H ′ 3 is*covered*by*a*single*cycle*in F i , then without loss of generality, we can assume the*cycle**passes**through*the (4,3) super-*edge*first. ...##
###
Minimum Reload Cost Cycle Cover in Complete Graphs
[article]

2017
*
arXiv
*
pre-print

Our algorithm also finds

arXiv:1706.05225v1
fatcat:xm3vmczg4fapxp7rq3er2zeztq
*a*monochromatic*cycle**cover*in complete*graphs**with**a*nearly equitable 2-*edge*-coloring except some special cases. ... Furthermore, we provide*a*polynomial-time algorithm that constructs*a*monochromatic*cycle**cover*in complete*graphs**with*an equitable 2-*edge*-coloring except possibly in*a*complete*graph**with*four vertices ...*A*Hamiltonian*cycle*of*a**graph*G is*a**cycle**passing**through*every vertex of G exactly once, and*a**graph*G containing*a*Hamiltonian*cycle*is called Hamiltonian. ...##
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An infinite family of tetravalent half-arc-transitive graphs

2006
*
Discrete Mathematics
*

*A*

*graph*is half-arc-transitive if its automorphism group acts transitively on vertices and

*edges*, but not on arcs. ... In this paper,

*a*new infinite family of tetravalent half-arc-transitive

*graphs*

*with*girth 4 is constructed, each of which has order 16m such that m > 1 is

*a*divisor of 2t 2 + 2t + 1 for

*a*positive integer ...

*passing*

*through*any

*given*vertex in V ( X). ...

##
###
Computing the Girth of a Planar Graph in $O(n \logn)$ Time

2010
*
SIAM Journal on Discrete Mathematics
*

We give an O(n log n) algorithm for computing the girth (shortest

doi:10.1137/090767868
fatcat:2vzo75g4izgyloewj3ljmvqxnq
*cycle*) of an undirected n-vertex planar*graph*. Our solution extends to any*graph*of bounded genus. ... It would be interesting to extend this algorithm to undirected*graphs**with*arbitrary positive real*edge*weights. It would also be interesting to find an o(n 3/2 ) algorithm for directed planar*graphs*. ... We have presented the fastest algorithm for computing the girth of an undirected planar*graph*and bounded genus*graph*. Our algorithm runs in O(n log n) time, improving the previous best algorithms. ...
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