Journal ArticleDOI
Partitioning Two-Coloured Complete Graphs into Two Monochromatic Cycles
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It is proved that there exists n0, such that, for every n≥n0 and every 2-colouring of the edges of the complete graph Kn, one can find two vertex-disjoint monochromatic cycles of different colours which cover all vertices of Kn.Abstract:
We prove that there exists n0, such that, for every n≥n0 and every 2-colouring of the edges of the complete graph Kn, one can find two vertex-disjoint monochromatic cycles of different colours which cover all vertices of Kn.read more
Citations
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Book ChapterDOI
The Regularity Lemma and Its Applications in Graph Theory
TL;DR: The present survey is a continuation of the earlier survey, and repeats (sometimes in a shortened form) parts of the first survey, but the emphasis is on new results.
Journal ArticleDOI
Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey
Mikio Kano,Xueliang Li +1 more
TL;DR: This paper surveys results on monochromatic and heterochromatics of an edge-colored graph and classifies the results into the following categories: vertex-partitions by monochrome subgraphs, such as cycles, paths, trees.
Journal ArticleDOI
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
TL;DR: It is shown that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyarfas.
Journal ArticleDOI
An improved bound for the monochromatic cycle partition number
TL;DR: Improving a result of Erdos, Gyarfas and Pyber for large n, it is shown that for every integer r>=2 there exists a constant n"0=n"0(r) such that if n>=n"-0 and the edges of the complete graph K"n are colored with r colors then the vertex set of K" n can be partitioned into at most 100rlogr vertex disjoint monochromatic cycles.
Journal ArticleDOI
Partitioning a graph into a cycle and an anticycle, a proof of Lehel's conjecture
Stéphane Bessy,Stéphan Thomassé +1 more
TL;DR: It is proved that every graph G has a vertex partition into a cycle and an anticycle (a cycle in the complement of G) and this problem was posed by Lehel and shown to be true for very large graphs by Luczak, Rodl and Szemeredi.
References
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Regular partitions of graphs
TL;DR: In this article, the authors generalize this result to arbitrary graphs, at the same time strengthening and simplifying the original bipartite result and showing that k-term arithmetic progression-free sets of integers must have density zero.
Szemeredi''s Regularity Lemma and its applications in graph theory
János Komlós,Miklós Simonovits +1 more
TL;DR: In this paper, the authors describe some typical applications and some generalizations of the regularity lemma, and also some new variants and generalizations appeared, and describe typical applications of the lemma.
Journal ArticleDOI
Blow-up Lemma
TL;DR: Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
Journal ArticleDOI
Vertex coverings by Monochromatic cycles and trees
TL;DR: If the edges of a finite complete graph K are colored with r colors then the vertex set of K can be covered by at most cr 2 log r vertex disjoint monochromatic cycles.
Journal ArticleDOI
Vertex coverings by monochromatic paths and cycles
TL;DR: The role of results on covering the vertices of 2-colored complete graphs by two paths or by two cycles of different color in determining path Ramsey numbers and in algorithms for finding long monochromatic paths or cycles in 2- colored complete graphs is shown.
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