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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

Carsten Thomassen
1999 Journal of combinatorial theory. Series B (Print)  
We prove 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.  ... 
doi:10.1006/jctb.1998.1868 fatcat:s5u3ggdzyrg6jbudvlughbix7e

Relaxed Two-Coloring of Cubic Graphs

Robert Berke, Tibor Szabó
2005 Discrete Mathematics & Theoretical Computer Science  
International audience We show 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.  ... 
doi:10.46298/dmtcs.3445 fatcat:dmvejlkcffbkxevuemb4vkclka

Relaxed two-coloring of cubic graphs

Robert Berke, Tibor Szabó
2007 Journal of combinatorial theory. Series B (Print)  
We show 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 .  ... 
doi:10.1016/j.jctb.2006.12.001 fatcat:r2cs4pn3yvc2rk4fzhlo4snuke

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Mikio Kano, Xueliang Li
2008 Graphs and Combinatorics  
Nowadays 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.  ... 
doi:10.1007/s00373-008-0789-5 fatcat:a4j64ccspze2td7cv3f247kf7u

WORM colorings

Wayne Goddard, Kirsti Wash, Honghai Xu
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.  ... 
doi:10.7151/dmgt.1814 fatcat:dozajhsrpzadvinmfnfcgp5joy

WORM colorings of planar graphs

Július Czap, Stanislav Jendrol', Juraj Valiska
2017 Discussiones Mathematicae Graph Theory  
Given three planar 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.  ... 
doi:10.7151/dmgt.1921 fatcat:h6snck4aifcejihhwpaeoqmkqm

Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

Zdeněk Dvořák, Daniel Král', Riste Škrekovski
2009 Journal of Graph Theory  
Finally, if G 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.  ... 
doi:10.1002/jgt.20414 fatcat:x7coyduqmnc6pksb4y7sgfwlbu

Coloring outerplanar graphs and planar 3-trees with small monochromatic components [article]

Michael A. Bekos, Carla Binucci, Michael Kaufmann, Chrysanthi Raftopoulou, Antonios Symvonis, Alessandra Tappini
2019 arXiv   pre-print
In this work, we continue 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.  ... 
arXiv:1911.10863v1 fatcat:4gtvbmax7fcazojq42yx4dgczi

Chords of 2-factors in planar cubic bridgeless graphs [article]

Ajit Diwan
2021 arXiv   pre-print
Another immediate consequence 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.  ... 
arXiv:2110.12584v1 fatcat:4t4nnchg3zfqzm3gwx3dke36ai

Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

Gábor Bacsó, Zsolt Tuza
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.  ... 
doi:10.46298/dmtcs.453 fatcat:seprmmtj7nh6fp55fy7klsccti

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

János Barát, Zoltán L. Blázsik, Gábor Damásdi
2021 arXiv   pre-print
Thomassen formulated 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-  ... 
arXiv:2108.08118v1 fatcat:77343cl63nbppjj6faarv5hfqi

Placing Regenerators in Optical Networks to Satisfy Multiple Sets of Requests

George B. Mertzios, Ignasi Sau, Mordechai Shalom, Shmuel Zaks
2012 IEEE/ACM Transactions on Networking  
We then study 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.  ... 
doi:10.1109/tnet.2012.2186462 fatcat:3hemod5t6zerjbu374xejiuh5q

Placing Regenerators in Optical Networks to Satisfy Multiple Sets of Requests [chapter]

George B. Mertzios, Ignasi Sau, Mordechai Shalom, Shmuel Zaks
2010 Lecture Notes in Computer Science  
We then study 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.  ... 
doi:10.1007/978-3-642-14162-1_28 fatcat:hmezgvjsq5cdfluu6ntvp353sm

Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

Hajo Broersma, Fedor V. Fomin, Jan Kratochvil, Gerhard J. Woeginger
2005 Algorithmica  
In particular, we prove 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.  ... 
doi:10.1007/s00453-005-1176-8 fatcat:feseqsxyr5h5ddrp3kqsvk6bxu

Spanning quadrangulations of triangulated surfaces

André Kündgen, Carsten Thomassen
2017 Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg  
We show 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.  ... 
doi:10.1007/s12188-016-0172-z fatcat:ubiuyfdxmvhihamq2yes2k6qeq
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