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### On the edge-colouring problem for unions of complete uniform hypergraphs

A.E. Brouwer, R. Tijdeman
1981 Discrete Mathematics
We now heave -2k+3~1-(k+\$n-l:~)sDs -&IL -I-1.) s k -2, and on the other hand D'= C n,(h-2)>--k;l (;)>y&n+2k+3. The edge-colouring problem for uni&as of complete hypergraphs Lemma 1.  ...  (Again we find b + 2c = g f 1 (moo h' and b -t 2c < j12, SO b +2c = g + The edge-co&wing probiem for unions of complete &re.rgmpk~ 12 2k + 1; but by (2j b + c =G k, which is impossible.)  ...

### Hypergraph Colouring and Degeneracy [article]

David R. Wood
2014 arXiv   pre-print
Moreover, the hypergraph is triangle-free, where a "triangle" in an r-uniform hypergraph consists of three edges whose union is a set of r+1 vertices.  ...  A hypergraph is "d-degenerate" if every subhypergraph has a vertex of degree at most d. A greedy algorithm colours every such hypergraph with at most d+1 colours.  ...  Acknowledgement Thanks to an anonymous referee for pointing out an error in an earlier version of this paper.  ...

### Ramsey numbers of sparse hypergraphs

David Conlon, Jacob Fox, Benny Sudakov
2009 Random structures & algorithms (Print)
Our methods also allow us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges. r(G) ≤ c(∆)n.  ...  We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree ∆ has Ramsey number at most c(∆, k)n, for an appropriate constant c(∆, k).  ...  For k-uniform hypergraphs H 1 , . . . , H q , the multicolour Ramsey number r(H 1 , . . . , H q ) is the minimum N such that, in any q-colouring of the edges of the complete k-uniform hypergraph K (k)  ...

### Ramsey numbers of sparse hypergraphs [article]

David Conlon, Jacob Fox, Benny Sudakov
2007 arXiv   pre-print
Our methods also allows us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges.  ...  We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree Δ has Ramsey number at most c(Δ, k)n, for an appropriate constant c(Δ, k).  ...  We would like to thank Jan Hladky for finding several typos in an earlier version of this paper.  ...

### Hedgehogs are not colour blind [article]

David Conlon, Jacob Fox, Vojtěch Rödl
2015 arXiv   pre-print
This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.  ...  We exhibit a family of 3-uniform hypergraphs with the property that their 2-colour Ramsey numbers grow polynomially in the number of vertices, while their 4-colour Ramsey numbers grow exponentially.  ...  One of the main outstanding problems in Ramsey theory is to decide whether the Ramsey number for complete 3-uniform hypergraphs is double exponential.  ...

### The NP-completeness of finding A-trails in Eulerian graphs and of finding spanning trees in hypergraphs

Lars Døvling Andersen, Herbert Fleischner
1995 Discrete Applied Mathematics
We prove, by a different reduction, that the problem remains NP-complete for simple, 3-connected, plane Eulerian graphs for which all face boundaries are 3-cycles or 4-cycles.  ...  We then apply this result to show that it is NP-complete to decide whether a linear hypergraph which is regular of degree 3 has a spanning tree.  ...  Acknowledgement The authors wish to thank Carsten Thomassen for his permission to include Theorem 4 and its proof, and to thank Steffen L. Lauritzen for helpful discussions.  ...

### Hedgehogs are not colour blind

David Conlon, Jacob Fox, Vojtěch Rödl
2017 Journal of Combinatorics
This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.  ...  We exhibit a family of 3-uniform hypergraphs with the property that their 2-colour Ramsey numbers grow polynomially in the number of vertices, while their 4-colour Ramsey numbers grow exponentially.  ...  One of the main outstanding problems in Ramsey theory is to decide whether the Ramsey number for complete 3-uniform hypergraphs is double exponential.  ...

### Loose Hamilton Cycles in Random 3-Uniform Hypergraphs [article]

Alan Frieze
2010 arXiv   pre-print
In the random hypergraph H=H(n,p;3) each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges x_i,y_i,x_i+1} for i=1,2,...,n/2.  ...  Acknowledgement I am grateful to Michael Krivelevich and Oleg Pikhurko for their comments.  ...  We say that a k-uniform sub-hypergraph C of H is a Hamilton cycle of type ℓ, for some 1 ≤ ℓ ≤ k, if there exists a cyclic ordering of the vertices V such that every edge consists of k consecutive vertices  ...

### Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

Alan Frieze
2010 Electronic Journal of Combinatorics
A loose Hamilton cycle can be described as a sequence of edges \$\{x_i,y_i,x_{i+1}\}\$ for \$i=1,2,\ldots,n/2\$ where \$x_1,x_2,\ldots,x_{n/2},y_1,y_2,\ldots,y_{n/2}\$ are all distinct.  ...  In the random hypergraph \$H=H_{n,p;3}\$ each possible triple appears independently with probability \$p\$.  ...  Acknowledgement I am grateful to Andrzej Dudek, Michael Krivelevich, Oleg Pikhurko and Andrzej Ruciński for their comments.  ...

### Monochromatic paths in 2-edge coloured graphs and hypergraphs [article]

Maya Stein
2022 arXiv   pre-print
Further, we give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete k-partite k-uniform hypergraph, and show that any 2-edge coloured bipartite graph has a partition  ...  We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours.  ...  For any n ∈ N, and any colouring of the edges of the 3-uniform complete hypergraph K n , there is a spanning bicolored tight path.  ...

### Quasi-Eulerian Hypergraphs

Amin Bahmanian, Mateja Šajna
2017 Electronic Journal of Combinatorics
We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in particular, we show that every 3-uniform hypergraph without cut edges admits an Euler  ...  triple systems, as well as recent results by Lonc and Naroski, who showed that the problem of existence of an Euler tour in a hypergraph is NP-complete.  ...  Acknowledgements The first author wishes to thank the Department of Mathematics and Statistics, University of Ottawa, for its hospitality during his postdoctoral fellowship, when this research was conducted  ...

### Page 6438 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews
For a finite hypergraph H, let P(A) be the number of proper vertex colourings of H in A colours.  ...  What is the smallest number of edges in a 3-chromatic 4-uniform hypergraph?  ...

### Near Perfect Coverings in Graphs and Hypergraphs

P. Frankl, V. Rödl
1985 European journal of combinatorics (Print)
Then there exists a collection of (I + 0(1))(2)/(2) induced subgraphs of gil on v vertices, isomorphic to CO and such that each edge (non-edge) of gil is covered by an edge (non-edge) of a graph in the  ...  Then there exists a collection of (I + 0(1)) x G)/(~) k-subsets of an n-set so that each r-subset is contained in at least one member of the collection.  ...  We denote by ;;(d (n, p) the random d-uniform hypergraph with each edge having probability p for being chosen. Also, K~ denotes the complete d-uniform hypergraph on n vertices. THEOREM 3.1.  ...

### Covering numbers for graphs and hypergraphs [article]

Bela Bollobas, Alexander Scott
2012 arXiv   pre-print
Consider the edge set E ′ consisting of edges with colour k + 1 or k + 2. Every vertex of H is incident with 1 or 2 edges from E ′ , so E ′ is a union of paths and cycles.  ...  Thus f (r, k) is the smallest d such that every d-uniform, r-regular hypergraph has a k-colouring in which every edge has a vertex of every colour.  ...

### Graph properties and hypergraph colourings

Jason I. Brown, Derek G. Corneil
1991 Discrete Mathematics
Given a graph property P, graph G and integer k 20, a P k-colouring of G is a function Jr:V(G)+ (1,. . ) k} such that the subgraph induced by each colour class has property P.  ...  In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic  ...  The order and size of a hypergraph H are IV(H)1 and [E(H)1 respectively, We remark that Abbott [l] used Ramsey's Theorem on edge colourings of complete hypergraphs to prove the existence of k-chromatic  ...
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