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

1981
*
Discrete Mathematics
*

We now heave -2k+3~1-(k+$n-l:~)sDs -&IL -I-1.) s k -2, and

doi:10.1016/0012-365x(81)90003-0
fatcat:jhqxurlm6zg5focfmcxogd3cyi
*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.) ...##
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Hypergraph Colouring and Degeneracy
[article]

2014
*
arXiv
*
pre-print

Moreover,

arXiv:1310.2972v3
fatcat:cl7ylos2a5b7fibfld4eaj2ixm
*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

2009
*
Random structures & algorithms (Print)
*

Our methods also allow us to prove quite sharp results

doi:10.1002/rsa.20260
fatcat:zyafoiymkba7poneyvqvl4zqmy
*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]

2007
*
arXiv
*
pre-print

Our methods also allows us to prove quite sharp results

arXiv:0710.0027v2
fatcat:t6ofknn4hzelxhcdnm3vbirc2e
*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]

2015
*
arXiv
*
pre-print

This is

arXiv:1511.00563v1
fatcat:q333shfe4neezdajhevrqmag74
*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

1995
*
Discrete Applied Mathematics
*

We prove, by a different reduction, that

doi:10.1016/0166-218x(95)80001-k
fatcat:ymtqpxxwvzdolgflhnumveuqx4
*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

2017
*
Journal of Combinatorics
*

This is

doi:10.4310/joc.2017.v8.n3.a4
fatcat:p2xzlfxvhvbcdbp2wu5hx2l6yy
*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]

2010
*
arXiv
*
pre-print

In

arXiv:1003.5817v1
fatcat:a7juufprtfh7rlyiranzdlnkcm
*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

2010
*
Electronic Journal of Combinatorics
*

A loose Hamilton cycle can be described as a sequence

doi:10.37236/477
fatcat:cgaeiux4qvbshldktghdwiwzay
*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]

2022
*
arXiv
*
pre-print

Further, we give a lower bound

arXiv:2204.12464v1
fatcat:wyiy5koju5e45ndvm4qmanl34u
*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

2017
*
Electronic Journal of Combinatorics
*

We provide necessary and sufficient conditions

doi:10.37236/6361
fatcat:4kqzraw3jvbo5ktmm773xwktz4
*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

1985
*
European journal of combinatorics (Print)
*

Then there exists a collection

doi:10.1016/s0195-6698(85)80045-7
fatcat:adqr2ty7qree7px3oefjykddei
*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]

2012
*
arXiv
*
pre-print

Consider

arXiv:1009.5893v2
fatcat:egxo65ohwndujo2rerek4affd4
*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

1991
*
Discrete Mathematics
*

Given a graph property P, graph G and integer k 20, a P k-

doi:10.1016/0012-365x(91)90034-y
fatcat:33nesuvxsbgshiupw2xrbcx4zi
*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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