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The number of cycles in 2-factors of cubic graphs

1990
*
Discrete Mathematics
*

*The*functions 5 can be viewed as

*the*average length

*of*

*cycles*

*in*

*2*-

*factors*

*of*

*the*extremal

*graphs*. f0 = lim inf IG,

*The*main results Let G and H be two disjoint

*cubic*3-connected

*graphs*. ... We denote by '9

*the*family

*of*all

*cubic*3-connected simple

*graphs*and by sr,

*the*subset

*of*planar

*graphs*

*in*5%

*The*

*number*

*of*vertices

*in*a

*graph*G is denoted by ICI. ...

*The*

*graph*G

*in*Fig. 8 ...

##
###
Klein Group And Four Color Theorem
[article]

2010
*
arXiv
*
pre-print

*In*this work methods

*of*construction

*of*

*cubic*

*graphs*are analyzed and a theorem

*of*existence

*of*a colored disc traversing each pair

*of*linked edges belonging to an elementary

*cycle*

*of*a planar

*cubic*

*graph*...

*The*

*number*

*of*edges

*in*a

*cubic*

*graph*is determined as: m = 3n /

*2*(1) Hence,

*the*

*number*

*of*edges

*in*such a

*graph*is always a multiple

*of*three. ... Since

*the*

*number*

*of*edges m is integer, therefore,

*the*

*number*

*of*vertices n

*in*an isomorphic

*cubic*

*graph*is even. ...

##
###
Switching 3-edge-colorings of cubic graphs
[article]

2021
*
arXiv
*
pre-print

Families

arXiv:2105.01363v1
fatcat:lozkxp6frzgeximswayntieygi
*of**cubic**graphs**of*orders 4n+*2*and 4n+4 with*2*^n edge-Kempe equivalence classes are presented; it is conjectured that there are no*cubic**graphs*with more edge-Kempe equivalence classes. ...*The*chromatic index*of*a*cubic**graph*is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings*of**cubic**graphs*. ...*The*observation at*the*end*of*Section*2*dates back to discussions with Petteri Kaski after*the*publication*of*[24] . ...##
###
Generation and properties of snarks

2013
*
Journal of combinatorial theory. Series B (Print)
*

*In*contrast to these positive results we also find counterexamples to eight previously published conjectures concerning

*cycle*coverings and

*the*general

*cycle*structure

*of*

*cubic*

*graphs*. ...

*In*

*the*second part

*of*

*the*paper we analyze

*the*sets

*of*generated snarks with respect to a

*number*

*of*properties and conjectures. ... Jan Goedgebeur is supported by a PhD grant

*of*

*the*Research Foundation

*of*Flanders (FWO). Jonas Hägglund is supported by

*the*National Graduate School

*in*Scientific Computing (NGSSC). ...

##
###
On S-packing edge-colorings of cubic graphs

2019
*
Discrete Applied Mathematics
*

Given a non-decreasing sequence S = (s 1 , s

doi:10.1016/j.dam.2018.12.035
fatcat:tb7yxo7uxfgfbf72norlhulwz4
*2*, . . . , s k )*of*positive integers, an Spacking edge-coloring*of*a*graph*G is a partition*of**the*edge set*of*G into k subsets {X 1 , X*2*, . . . , X k } ... Among other results, we prove that*cubic**graphs*having a*2*-*factor*are (1, 1, 1, 3 , 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4) -packing edgecolorable and (1, 1,*2*,*2*,*2*,*2*,*2*)-packing edge-colorable ... For a*cubic**graph*G having a*2*-*factor*,*the*oddness*of*G is*the*minimum*number**of*odd*cycle*among all*2*-*factors**of*G. According to Petersen's theorem, every bridgeless*cubic**graph*has a*2*-*factor*. ...##
###
On S-packing edge-colorings of cubic graphs
[article]

2017
*
arXiv
*
pre-print

Given a non-decreasing sequence S = (s 1,s

arXiv:1711.10906v1
fatcat:kih3n7mfdrathfqa5mwarpibkq
*2*,. .. ,s k)*of*positive integers, an S-packing edge-coloring*of*a*graph*G is a partition*of**the*edge set*of*G into k subsets X 1 ,X*2*,. .. ... Among other results, we prove that*cubic**graphs*having a*2*-*factor*are (1,1,1,3,3)-packing edge-colorable, (1,1,1,4,4,4,4,4)-packing edge-colorable and (1,1,2,2,2,2,2)-packing edge-colorable. ... For a*cubic**graph*G having a*2*-*factor*,*the*oddness*of*G is*the*minimum*number**of*odd*cycle*among all*2*-*factors**of*G. According to Petersen's theorem, every bridgeless*cubic**graph*has a*2*-*factor*. ...##
###
Improved Approximations for Cubic and Cubic Bipartite TSP
[article]

2016
*
arXiv
*
pre-print

For

arXiv:1507.07121v3
fatcat:5fnzhlfacbda7ep7ft7ei3b42m
*2*-connected*cubic**graphs*, we show that*the*techniques*of*Moemke and Svensson (2011) can be combined with*the*techniques*of*Correa, Larre and Soto (2012), to obtain a tour*of*length at most (4/3-1/8754 ... We show improved approximation guarantees for*the*traveling salesman problem on*cubic**graphs*, and*cubic*bipartite*graphs*. ... for suggesting*the*simplified proof for*the*result*in*Section 3 for*cubic*non-bipartite*graphs*. ...##
###
Improved Approximations for Cubic Bipartite and Cubic TSP
[chapter]

2016
*
Lecture Notes in Computer Science
*

For

doi:10.1007/978-3-319-33461-5_21
fatcat:2enha5j7zrhine64muvd467gci
*2*-connected*cubic**graphs*, we show that*the*techniques*of*Mömke and Svensson can be combined with*the*techniques*of*Correa, Larré and Soto, to obtain a tour*of*length at most (4/3 − 1/8754)n. ... We show improved approximation guarantees for*the*traveling salesman problem on*cubic*bipartite*graphs*and*cubic**graphs*. ... for suggesting*the*simplified proof for*the*result*in*Section 3 for*cubic*non-bipartite*graphs*. ...##
###
On snarks that are far from being 3-edge colorable
[article]

2012
*
arXiv
*
pre-print

Furthermore

arXiv:1203.2015v1
fatcat:a4ugnhjayjhypg44v5ayrv3egi
*the*counterexample presented has*the*interesting property that no*2*-*factor*can be part*of*a*cycle*double cover. ...*In*this note we construct two infinite snark families which have high oddness and low circumference compared to*the**number**of*vertices. ...*The*oddness*of*a bridgeless*cubic**graph*G is defined as*the*minimum*number**of*odd components*in*any*2*-*factor**in*G and is denoted by o(G). ...##
###
Shortest cycle covers and cycle double covers with large 2-regular subgraphs

2013
*
Journal of Combinatorics
*

*In*this paper, we show that many snarks have a shortest

*cycle*cover

*of*length 4 3 m + c for a constant c, where m is

*the*

*number*

*of*edges

*in*

*the*

*graph*,

*in*agreement with

*the*conjecture that all snarks have ...

*In*particular, we prove that

*graphs*with perfect matching index at most 4 have

*cycle*covers

*of*length 4 3 m and satisfy

*the*(1,

*2*)covering conjecture

*of*Zhang, and that

*graphs*with large circumference ... Acknowledgments

*The*authors would like to thank

*the*referees for their constructive criticism and Lars-DanielÖhman for his comments on

*the*manuscript. ...

##
###
Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks
[article]

2014
*
arXiv
*
pre-print

*The*

*graphs*we construct are snarks so we get

*the*same upper bound for

*the*shortness coefficient

*of*snarks, and we prove that

*the*constructed

*graphs*have an oddness growing linearly with

*the*

*number*

*of*vertices ... We present a construction which shows that there is an infinite set

*of*cyclically 4-edge connected

*cubic*

*graphs*on n vertices with no

*cycle*longer than c_4 n for c_4=12/13, and at

*the*same time prove that ... One measure

*of*

*the*degree

*of*non-colourability for a

*cubic*

*graph*is its oddness, i.e.

*the*minimum

*number*

*of*odd

*cycles*

*in*any

*2*-

*factor*

*of*

*the*

*graph*. ...

##
###
The Color Number of Cubic Graphs Having a Spanning Tree with a Bounded Number of Leaves

2021
*
Theory and Applications of Graphs
*

*In*this paper, we extend these observations by obtaining a bound for

*the*color

*number*

*of*

*cubic*

*graphs*having a spanning tree with a bounded

*number*

*of*leaves. ...

*The*color

*number*c(G)

*of*a

*cubic*

*graph*G is

*the*minimum cardinality

*of*a color class

*of*a proper 4-edge-coloring

*of*G. ...

*The*oddness

*of*a

*cubic*

*graph*G, denoted by ω(G), is

*the*smallest

*number*

*of*odd

*cycles*

*in*a

*2*-

*factor*

*of*G, where a

*2*-

*factor*is a spanning subgraph

*in*which every vertex has degree

*2*. ...

##
###
Shortest cycle covers and cycle double covers with large 2-regular subgraphs
[article]

2013
*
arXiv
*
pre-print

*In*this paper we show that many snarks have shortest

*cycle*covers

*of*length 4/3m+c for a constant c, where m is

*the*

*number*

*of*edges

*in*

*the*

*graph*,

*in*agreement with

*the*conjecture that all snarks have shortest ...

*In*particular we prove that

*graphs*with perfect matching index at most 4 have

*cycle*covers

*of*length 4/3m and satisfy

*the*(1,2)-covering conjecture

*of*Zhang, and that

*graphs*with large circumference have ... Short covers

*in*

*graphs*with Oddness

*2*Recall that

*the*oddness o(G)

*of*a

*cubic*

*graph*G is

*the*minimum

*number*

*of*odd

*cycles*

*in*any

*2*-

*factor*

*of*G. ...

##
###
Tutte's 5-flow conjecture for highly cyclically connected cubic graphs

2010
*
Discrete Mathematics
*

*In*1954, Tutte conjectured that every bridgeless

*graph*has a nowhere-zero 5-flow. Let ω be

*the*minimum

*number*

*of*odd

*cycles*

*in*a

*2*-

*factor*

*of*a bridgeless

*cubic*

*graph*. ... We show that if a

*cubic*

*graph*G has no edge cut with fewer than 5/

*2*ω - 1 edges that separates two odd

*cycles*

*of*a minimum

*2*-

*factor*

*of*G, then G has a nowhere-zero 5-flow. ... By Petersen's theorem, every bridgeless

*cubic*

*graph*G has a

*2*-

*factor*and

*the*oddness ω(G) is

*the*minimum

*number*

*of*odd

*cycles*

*in*a

*2*-

*factor*

*of*G. ...

##
###
Weak oddness as an approximation of oddness and resistance in cubic graphs
[article]

2016
*
arXiv
*
pre-print

We introduce weak oddness ω_ w, a new measure

arXiv:1602.02949v1
fatcat:x7u2bjarszbnjjoggwytk2osr4
*of*uncolourability*of**cubic**graphs*, defined as*the*least*number**of*odd components*in*an even*factor*. ... For every bridgeless*cubic**graph*G, ρ(G)<ω_ w(G)<ω(G), where ρ(G) denotes*the*resistance*of*G and ω(G) denotes*the*oddness*of*G, so this new measure is an approximation*of*both oddness and resistance. ... We would like to thank Barbora Candráková, Edita Máčajová, Eckhard Steffen, and Martin Škoviera for many fruitful discussions on topics related to even*factors**in**cubic**graphs*. ...
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