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Two-Coloring the Edges of a Cubic Graph Such That Each Monochromatic Component Is a Path of Length at Most 5

1999
*
Journal of combinatorial theory. Series B (Print)
*

We prove

doi:10.1006/jctb.1998.1868
fatcat:s5u3ggdzyrg6jbudvlughbix7e
*the*conjecture made by Bermond, Fouquet, Habib, and Pe roche in 1984*that*every*cubic**graph*has an*edge*-*coloring*as described in*the*title.*The*number*5*cannot be replaced by 4. ... Then*the**edge*set*of*H has*a**coloring*in*two**colors*and an orientation*of**the**edges**such**that**each**monochromatic**component**is**a*directed*path**of**length**at**most**5*. ... We*color**the**edges**of*G i in red and blue and we orient*the**edges**such**that**each**monochromatic**component**is**a*directed*path**of**length**at**most**5*. ...##
###
Relaxed Two-Coloring of Cubic Graphs

2005
*
Discrete Mathematics & Theoretical Computer Science
*

International audience We show

doi:10.46298/dmtcs.3445
fatcat:dmvejlkcffbkxevuemb4vkclka
*that*any*graph**of*maximum degree*at**most*$3$ has*a**two*-*coloring*,*such**that*one*color*-class*is*an independent set while*the*other*color*induces*monochromatic**components**of*... On*the*other hand for any constant $C$ we exhibit*a*$4$-regular*graph*,*such**that**the*deletion*of*any independent set leaves*at*least one*component**of*order greater than $C$. ... We are indebted to Péter Mihók for calling our attention to*the*asymmetric version*of*relaxed*colorings*. ...##
###
Relaxed two-coloring of cubic graphs

2007
*
Journal of combinatorial theory. Series B (Print)
*

We show

doi:10.1016/j.jctb.2006.12.001
fatcat:r2cs4pn3yvc2rk4fzhlo4snuke
*that*any*graph**of*maximum degree*at**most*3 has*a**two*-*coloring**such**that*one*color*-class*is*an independent set, while*the*other*color*-class induces*monochromatic**components**of*order*at**most*750. ... On*the*other hand, for any constant C, we exhibit*a*4-regular*graph**such**that**the*deletion*of*any independent set leaves*at*least one*component**of*order greater than C. ... Acknowledgments We are indebted to Péter Mihók for calling our attention to*the*asymmetric version*of*relaxed*colorings*. We thank Yoshio Okamoto for*the**graph*on Fig. 7 . ...##
###
Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

2008
*
Graphs and Combinatorics
*

Nowadays

doi:10.1007/s00373-008-0789-5
fatcat:a4j64ccspze2td7cv3f247kf7u
*the*term*monochromatic*and heterochromatic (or rainbow, multicolored) subgraphs*of*an*edge**colored**graph*appeared frequently in literature, and many results on this topic have been obtained. ... We have to point out*that*there are*a*lot*of*results*of*Ramsey type problem on*monochromatic*and heterochromatic subgraphs. ... In 1984, Bermond, Fouquet, Habib and Péroche [11] conjectured*that*every*cubic**graph*has*a*2-*edge**coloring**such**that**each**monochromatic**component**is**a**path**of**length**at**most**5*. ...##
###
WORM colorings

2015
*
Discussiones Mathematicae Graph Theory
*

*The*focus

*is*on

*the*case

*that*F

*is*

*the*

*path*on three vertices. ... Given

*a*

*graph*G and

*a*

*graph*F , we define an F -WORM

*coloring*

*of*G as

*a*

*coloring*

*of*

*the*vertices

*of*G without

*a*rainbow or

*monochromatic*subgraph H isomorphic to F . ... Recall

*that*

*a*1-defective 2-

*coloring*

*of*

*a*

*graph*G

*is*

*a*2-

*coloring*

*such*

*that*

*each*vertex has

*at*

*most*one neighbor

*of*its

*color*. ...

##
###
WORM colorings of planar graphs

2017
*
Discussiones Mathematicae Graph Theory
*

Given three planar

doi:10.7151/dmgt.1921
fatcat:h6snck4aifcejihhwpaeoqmkqm
*graphs*... Acknowledgement*The*authors would like to express their sincere gratitude to*the*referee for*the*insightful comments and valuable suggestions, which led to*a*substantial improvement*of**the*original version ...*of*this paper. ...##
###
Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

2009
*
Journal of Graph Theory
*

Finally, if G

doi:10.1002/jgt.20414
fatcat:x7coyduqmnc6pksb4y7sgfwlbu
*is**5*-connected, then*the*number*of**colors**is**at**most*25 58 n − 22 29 . ... If G*is**a*3-connected plane*graph*with n vertices, then*the*number*of**colors*in*such**a**coloring*does not exceed 7n−8 9 . ... We would also like to thank one*of**the*referees for pointing out an error in*the*original proof*of*Theorem*5*. ...##
###
Coloring outerplanar graphs and planar 3-trees with small monochromatic components
[article]

2019
*
arXiv
*
pre-print

In this work, we continue

arXiv:1911.10863v1
fatcat:4gtvbmax7fcazojq42yx4dgczi
*the*study*of*vertex*colorings**of**graphs*, in which adjacent vertices are allowed to be*of**the*same*color*as long as*each**monochromatic*connected*component**is**of*relatively small ... We focus on*colorings*with*two*and three available*colors*and present improved bounds on*the*size*of**the**monochromatic*connected*components*for*two*meaningful subclasses*of*planar*graphs*, namely maximal ... Lovász [20] proved*that*any (not necessarily planar)*cubic**graph*G admits*a*2-*coloring*in which all*monochromatic**components*are*of*size*at**most**two*, i.e., mcc 2 (G) ≤ 2. ...##
###
Chords of 2-factors in planar cubic bridgeless graphs
[article]

2021
*
arXiv
*
pre-print

Another immediate consequence

arXiv:2110.12584v1
fatcat:4t4nnchg3zfqzm3gwx3dke36ai
*of**the*main result*is**that*for any*two**edges*contained in*a*facial cycle*of**a*2-*edge*-connected planar*cubic**graph*, there exists*a*2-factor in*the**graph**such**that*both*edges*... We show*that*every*edge*in*a*2-*edge*-connected planar*cubic**graph**is*either contained in*a*2-*edge*-cut or*is**a*chord*of*some cycle*that**is*contained in*a*2-factor*of**the**graph*. ... An*edge*in*a**cubic**graph**is**a*chord*of*some cycle iff it*is*not contained in an*edge*-cut*of*size*at**most**two*. ...##
###
Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009
*
Discrete Mathematics & Theoretical Computer Science
*

*Graphs*and Algorithms International audience

*A*clique-transversal set in

*a*

*graph*

*is*

*a*subset

*of*

*the*vertices

*that*meets all maximal complete subgraphs on

*at*least

*two*vertices. ... We also prove

*that*

*the*vertex set

*of*any connected claw-free

*graph*

*of*maximum degree

*at*

*most*four, other than an odd cycle longer than three, can be partitioned into

*two*clique-transversal sets. ...

*The*proof can be done in

*two*steps: • Given

*a*

*cubic*

*graph*G = (V, E), replace

*each*

*edge*e = xy ∈ E by

*a*

*path*xv e w e y

*of*

*length*three. ...

##
###
Crumby colorings – red-blue vertex partition of subcubic graphs regarding a conjecture of Thomassen
[article]

2021
*
arXiv
*
pre-print

Thomassen formulated

arXiv:2108.08118v1
fatcat:77343cl63nbppjj6faarv5hfqi
*the*following conjecture: Every 3-connected*cubic**graph*has*a*red-blue vertex*coloring**such**that**the*blue subgraph has maximum degree*at**most*1 (*that**is*, it consists*of**a*matching ... Since all*monochromatic**components*are small in this*coloring*and there*is**a*certain irregularity, we call*such**a**coloring*crumby. ... Acknowledgements*The*first author would like to thank Bjarne Toft for 25 years*of*friendship and encouragement. He influenced our work in Section 4.*The*first author was partially supported by ERC Ad- ...##
###
Placing Regenerators in Optical Networks to Satisfy Multiple Sets of Requests

2012
*
IEEE/ACM Transactions on Networking
*

We then study

doi:10.1109/tnet.2012.2186462
fatcat:3hemod5t6zerjbu374xejiuh5q
*the*case where G*is**a**path*, proving*that**the*problem*is*NP-hard for any d, p ≥ 2, even if there are*two**edges**of**the**path**such**that*any lightpath uses*at*least one*of*them. ... Interestingly, we show*that**the*problem*is*polynomial-time solvable in*paths*when all*the*lightpaths share*the*first*edge**of**the**path*, as well as when*the*number*of*lightpaths sharing an*edge**is*bounded ... Appeared in Proceedings*of**the*37th International Colloquium on Automata, Languages and Programming (ICALP), Bordeaux, France, July 2010, Volume 2, pp. 333-344. Best paper award*of*Track C. ...##
###
Placing Regenerators in Optical Networks to Satisfy Multiple Sets of Requests
[chapter]

2010
*
Lecture Notes in Computer Science
*

We then study

doi:10.1007/978-3-642-14162-1_28
fatcat:hmezgvjsq5cdfluu6ntvp353sm
*the*case where G*is**a**path*, proving*that**the*problem*is*NP-hard for any d, p ≥ 2, even if there are*two**edges**of**the**path**such**that*any lightpath uses*at*least one*of*them. ... Interestingly, we show*that**the*problem*is*polynomial-time solvable in*paths*when all*the*lightpaths share*the*first*edge**of**the**path*, as well as when*the*number*of*lightpaths sharing an*edge**is*bounded ... Appeared in Proceedings*of**the*37th International Colloquium on Automata, Languages and Programming (ICALP), Bordeaux, France, July 2010, Volume 2, pp. 333-344. Best paper award*of*Track C. ...##
###
Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

2005
*
Algorithmica
*

In particular, we prove

doi:10.1007/s00453-005-1176-8
fatcat:feseqsxyr5h5ddrp3kqsvk6bxu
*that*it*is*NP-complete to decide if*a*planar*graph*can be 2-*colored*so*that*no cycle*of**length**at**most**5**is**monochromatic*. ... Abstract We consider*the*problem*of**coloring**a*planar*graph*with*the*minimum number*of**colors*so*that**each**color*class avoids one or more forbidden*graphs*as subgraphs. ... Acknowledgments We are grateful to Oleg Borodin, Alesha Glebov, Sasha Kostochka, and Carsten Thomassen for fruitful discussions on*the*topic*of*this paper. ...##
###
Spanning quadrangulations of triangulated surfaces

2017
*
Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
*

We show

doi:10.1007/s12188-016-0172-z
fatcat:ubiuyfdxmvhihamq2yes2k6qeq
*that**the*dual*graph**of*an Eulerian triangulation*of*an orientable surface other than*the*sphere has*a*perfect matching M and an M -alternating noncontractible cycle. ... We observe*that*4-vertex-*colorability**of**a*triangulation on*a*surface can be expressed in terms*of*spanninq quadrangulations, and we establish connections between spanning quadrangulations and cycles in ... We start our investigation with*the*observation*that*if we have*a*planar*graph*in which*at**most*one face has odd*length*, then in fact every face has even*length*and*the**graph**is*bipartite. ...
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