Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

Letzter, Shoham

doi:10.37236/7239

Cited by 17 publications

(18 citation statements)

References 22 publications

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“…Furthermore, Example 3.3 shows that this is best possible if true. Regarding the distinct colors variant of these problems, we make the following conjecture which is true for r = 1 as above, and for r = 2 by Letzter's result [26]. Furthermore, Example 3.5 shows that this is best possible if true.…”

Section: Open Problemsmentioning

confidence: 75%

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Partitioning Random Graphs into Monochromatic Components

Bal

DeBiasio

2017

Electron. J. Combin.

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Erdős, Gyárfás, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r − 1 monochromatic components; this is a strengthening of a conjecture of Lovász (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into r monochromatic components is possible for sufficiently large r-colored complete graphs.We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if p ≥ 27 log n n 1/3 , then a.a.s. in every 2coloring of G(n, p) there exists a partition into two monochromatic components, and for r ≥ 2 if p r log n n 1/r , then a.a.s. there exists an r-coloring of G(n, p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gyárfás (1977) about large monochromatic components in r-colored complete graphs. We show that if p = ω(1) n , then a.a.s. in every r-coloring of G(n, p) there exists a monochromatic component of order at least (1 − o(1)) n r−1 .

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Section: Open Problemsmentioning

confidence: 75%

“…That is, if δ(G) > 3n/4, then cp 2 (G) ≤ 2 (with cycles of different colors); they also provided an example which shows that the conjecture would be best possible. DeBiasio and Nelsen [8] proved that this holds for G with δ(G) > (3/4 + o(1))n and then Letzter [26] proved that it holds exactly for sufficiently large n.…”

Section: Large Minimum Degreementioning

confidence: 96%

Partitioning Random Graphs into Monochromatic Components

Bal

DeBiasio

2017

Electron. J. Combin.

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“…Some generalizations of these results concerning more complicated graphs other than paths or cycles were obtained in [8,21]. Similar properties of host graphs other than complete graphs were also studied: complete bipartite graphs are considered in [9,13,17], complete graphs with only few edges missing in [11], graphs with large minimum degree in [2,5,18] and graphs with small independence number in [20]. For further results and research directions we refer the reader to the recent survey by Gyárfás [10].…”

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confidence: 86%

Monochromatic cycle covers in random graphs

Korándi

Mousset

Nenadov

et al. 2018

Random Struct Algorithms

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A classic result of Erdős, Gyárfás and Pyber states that for every coloring of the edges of Kn with r colors, there is a cover of its vertex set by at most f(r)=O(r2logr) vertex‐disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of Kn but only on the number of colors. We initiate the study of this phenomenon in the case where Kn is replaced by the random graph G(n,p). Given a fixed integer r and p=p(n)≥n−1/r+ε, we show that with high probability the random graph G∼G(n,p) has the property that for every r‐coloring of the edges of G, there is a collection of f′(r)=O(r8logr) monochromatic cycles covering all the vertices of G. Our bound on p is close to optimal in the following sense: if p≪(logn/n)1/r, then with high probability there are colorings of G∼G(n,p) such that the number of monochromatic cycles needed to cover all vertices of G grows with n.

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“…Most of the difficulty is in the proof for r = 3, but we include a short proof for r = 2 for completeness. Actually, the r = 2 case (for n large) already follows from a difficult result of Letzter [14], who showed that when ≥ ∕ δ G n ( ) 3 4, the vertices can be partitioned into two monochromatic cycles of different colors, for every 2-coloring of G. Before turning to the proofs, we mention the following construction of Bal and DeBiasio [3], which shows that the minimum degree condition in Conjecture 1.5 cannot be improved. F I G U R E 1 An illustration of Example 3.1 for t = 2 and t = 3 (here gray represents red and black represents blue) 2 In fact, we need to be a bit more careful here.…”

Section: Of Distinct Colorsmentioning

confidence: 99%

Partitioning a graph into monochromatic connected subgraphs

Girão

Letzter

Sahasrabudhe

2018

Journal of Graph Theory

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We show that every 2‐edge‐colored graph on n vertices with minimum degree at least 2 n − 5 3 can be partitioned into two monochromatic connected subgraphs, provided n is sufficiently large. This minimum degree condition is tight and the result proves a conjecture of Bal and DeBiasio. We also make progress on another conjecture of Bal and DeBiasio on covering graphs with large minimum degree with monochromatic components of distinct colors.

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Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

Cited by 17 publications

References 22 publications

Partitioning Random Graphs into Monochromatic Components

Partitioning Random Graphs into Monochromatic Components

Monochromatic cycle covers in random graphs

Partitioning a graph into monochromatic connected subgraphs

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