## Rainbow hamilton cycles in random regular graphs

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Venue: | In preparation |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Janson_rainbowhamilton,

author = {Svante Janson and Nicholas Wormald},

title = {Rainbow hamilton cycles in random regular graphs},

booktitle = {In preparation},

year = {}

}

### OpenURL

### Abstract

Abstract. A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, for fixed d ≥ 8. 1.

### Citations

1791 | Random Graphs
- Bollobás
- 1985
(Show Context)
Citation Context ...to extend the study to multigraphs. Recall that a convenient way (at least for theoretical purposes) to generate a random regular graph is the so-called configuration model or pairing model, see e.g. =-=[1]-=- or [19]: We start with nd points partitioned into n cells of d points each. We then take a random pairing of the points into nd/2 pairs (assuming nd to be even). Collapsing each cell to a vertex and ... |

728 | On triangle-free random graphs - Luczak |

81 | Almost all regular graphs are hamiltonian, Random Structures and Algorithms 5
- Robinson, Wormald
- 1994
(Show Context)
Citation Context ...5, 05C45, 60C05). The second author acknowledges the support of the Canadian Research Chairs Program and NSERC. 1s2 SVANTE JANSON AND NICHOLAS WORMALD Recall that it was shown by Robinson and Wormald =-=[16, 17]-=- that G(n, d) whp contains a Hamilton cycle as soon as d ≥ 3. In our setting, when d = 2q has to be even, we thus whp have Hamilton cycles, ignoring the colouring, when d ≥ 4, but rainbow Hamilton cyc... |

58 |
N.C.: Almost all cubic graphs are hamiltonian
- Robinson, Wormald
(Show Context)
Citation Context ...5, 05C45, 60C05). The second author acknowledges the support of the Canadian Research Chairs Program and NSERC. 1s2 SVANTE JANSON AND NICHOLAS WORMALD Recall that it was shown by Robinson and Wormald =-=[16, 17]-=- that G(n, d) whp contains a Hamilton cycle as soon as d ≥ 3. In our setting, when d = 2q has to be even, we thus whp have Hamilton cycles, ignoring the colouring, when d ≥ 4, but rainbow Hamilton cyc... |

48 |
Models of random regular graphs. Surveys in combinatorics
- Wormald
- 1999
(Show Context)
Citation Context ...er tends to infinity) as an open problem. The proof of Theorem 1.1 is based on the small subgraph conditioning method introduced by Robinson and Wormald [16, 17], and further developed in [12], [15], =-=[19]-=- and [13, Chapter 9]. However, for this problem we have to consider the colourings of the small subgraphs too, see Section 3. Acknowledgements. This problem was suggested by Alan Frieze during the Con... |

37 | Random regular graphs: asymptotic distributions and contiguity
- Janson
(Show Context)
Citation Context ...xpected number tends to infinity) as an open problem. The proof of Theorem 1.1 is based on the small subgraph conditioning method introduced by Robinson and Wormald [16, 17], and further developed in =-=[12]-=-, [15], [19] and [13, Chapter 9]. However, for this problem we have to consider the colourings of the small subgraphs too, see Section 3. Acknowledgements. This problem was suggested by Alan Frieze du... |

30 | Generating and counting hamilton cycles in random regular graphs
- Frieze, Jerrum, et al.
- 1996
(Show Context)
Citation Context ...problems, in particular in the theory of random regular graphs, see e.g. [13, Chapter 9], [19] and [11]. (For applications to random hypergraphs, see [5, 4].) As often pointed out by Alan Frieze, see =-=[4, 5, 6]-=-, the method can be regarded as an analysis of variance. The main idea is that we consider some random variable, Y say, that counts occurrences of some structure, and let a parameter n → ∞. Typically,... |

27 | N.: Random matchings which induce hamilton cycles and hamiltonian decompositions of random regular graphs
- Kim, Wormald
(Show Context)
Citation Context ...th o( √ n) randomly specified edges whp has a Hamilton cycle passing through all the specified edges (and, moreover, in randomly prespecified directions). It has further been shown by Kim and Wormald =-=[14]-=- that a random 2q-regular graph whp has an edgedecomposition into q Hamilton cycles, provided q ≥ 2. It is natural to ask whether, similarly, a randomly coloured 2q-regular graph with n colours and q ... |

20 |
The number of matchings in random regular graphs and bipartite graphs
- Bollobás, McKay
- 1986
(Show Context)
Citation Context ...easily checked that, as n → ∞, E Z → 0 for d ≤ 6, while E Z → ∞ for d ≥ 7. In particular, for d ≤ 6 there is whp no rainbow perfect matching. Furthermore, for d ≥ 7, an argument similar to the one in =-=[2]-=- (and much simpler than the proof of Lemma 3.4 above, since we only need to maximize over one variable) yields E(Z2 ) → (E Z) 2 d − 1 � . d(d − 3) Finally, defining Xij as before, Theorem 3.1 applies ... |

9 | Perfect Matchings in Random s-uniform Hypergraphs
- Frieze, Janson
- 1995
(Show Context)
Citation Context ...[16, 17] has been successfully applied to several problems, in particular in the theory of random regular graphs, see e.g. [13, Chapter 9], [19] and [11]. (For applications to random hypergraphs, see =-=[5, 4]-=-.) As often pointed out by Alan Frieze, see [4, 5, 6], the method can be regarded as an analysis of variance. The main idea is that we consider some random variable, Y say, that counts occurrences of ... |

8 | Perfect matchings in random r-regular, s-uniform hypergraphs
- Cooper, Frieze, et al.
- 1996
(Show Context)
Citation Context ...[16, 17] has been successfully applied to several problems, in particular in the theory of random regular graphs, see e.g. [13, Chapter 9], [19] and [11]. (For applications to random hypergraphs, see =-=[5, 4]-=-.) As often pointed out by Alan Frieze, see [4, 5, 6], the method can be regarded as an analysis of variance. The main idea is that we consider some random variable, Y say, that counts occurrences of ... |

7 | Permutation pseudographs and contiguity
- Greenhill, Janson, et al.
(Show Context)
Citation Context ...itioning method introduced by Robinson and Wormald [16, 17] has been successfully applied to several problems, in particular in the theory of random regular graphs, see e.g. [13, Chapter 9], [19] and =-=[11]-=-. (For applications to random hypergraphs, see [5, 4].) As often pointed out by Alan Frieze, see [4, 5, 6], the method can be regarded as an analysis of variance. The main idea is that we consider som... |

5 | cycles containing randomly selected edges in random regular graphs
- Robinson, Wormald
(Show Context)
Citation Context ...ton cycle manages to pick up an edge of each colour in a random 8-regular graph, when there are only four edges of each colour to choose from. Remark 1.2. In a similar direction, Robinson and Wormald =-=[18]-=- showed that a random 3-regular graph with o( √ n) randomly specified edges whp has a Hamilton cycle passing through all the specified edges (and, moreover, in randomly prespecified directions). It ha... |

4 | Poisson-Dirichlet distribution for random Belyi surfaces
- Gamburd
- 2006
(Show Context)
Citation Context ...ng switches) have connection graphs that are stars. In [10], this was extended to graphs where a random subset of the vertices have a star as connection graph and the rest the complete graph. Gamburd =-=[7]-=- studied long cycles in random oriented cubic graphs; here the connection graph is a directed 3-cycle at each vertex. 3. Small subgraphs The small subgraph conditioning method introduced by Robinson a... |

4 |
Asymptotic properties of the connectivity number of random railways
- Garmo
- 1999
(Show Context)
Citation Context ...ictions) for one side of the bipartition, and a matching with d/2 edges for the other side. We do not know of any general study, but a few examples of this type have appeared in the literature: Garmo =-=[8, 9]-=- studied random railways; these are regular (typically cubic) graphs where the vertices (representing switches) have connection graphs that are stars. In [10], this was extended to graphs where a rand... |

3 |
Multi-coloured Hamilton cycles in randomly coloured random graphs
- Cooper, Frieze
- 2002
(Show Context)
Citation Context ... randomly coloured random graph by Gc(n, 2q). Our main result is the following on randomly coloured random regular graphs. (For some related results on the random graph G(n, m), see Cooper and Frieze =-=[3]-=-.) We say that an event holds with high probability (whp), if it holds with probability tending to 1 as n → ∞. (All unspecified limits in this paper are for n → ∞.) Theorem 1.1. Consider the randomly ... |

3 |
Random railways modeled as random 3-regular graphs
- Garmo
- 1996
(Show Context)
Citation Context ...ictions) for one side of the bipartition, and a matching with d/2 edges for the other side. We do not know of any general study, but a few examples of this type have appeared in the literature: Garmo =-=[8, 9]-=- studied random railways; these are regular (typically cubic) graphs where the vertices (representing switches) have connection graphs that are stars. In [10], this was extended to graphs where a rand... |

1 | On generalized random railways
- Garmo, Janson, et al.
(Show Context)
Citation Context ... have appeared in the literature: Garmo [8, 9] studied random railways; these are regular (typically cubic) graphs where the vertices (representing switches) have connection graphs that are stars. In =-=[10]-=-, this was extended to graphs where a random subset of the vertices have a star as connection graph and the rest the complete graph. Gamburd [7] studied long cycles in random oriented cubic graphs; he... |

1 |
of random regular graphs. Random Strucures Algorithms 10
- Molloy, Robalewska, et al.
(Show Context)
Citation Context ...d number tends to infinity) as an open problem. The proof of Theorem 1.1 is based on the small subgraph conditioning method introduced by Robinson and Wormald [16, 17], and further developed in [12], =-=[15]-=-, [19] and [13, Chapter 9]. However, for this problem we have to consider the colourings of the small subgraphs too, see Section 3. Acknowledgements. This problem was suggested by Alan Frieze during t... |