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Proof of a Conjecture of Bollobás and Eldridge for Graphs of Maximum Degree Three*

Béla Csaba†‡, Ali Shokoufandeh‡, Endre Szemerédi
2003 Combinatorica  
A conjecture of Bollob as and Eldridge 5] asserts that, if ( (G 1 ) + 1)( (G 2 ) + 1) n + 1 then there is a packing of G 1 and G 2 . We prove this conjecture when (G 1 ) = 3, for su ciently large n.  ...  Let G 1 and G 2 be graphs on n vertices. If there are edge-disjoint copies of G 1 and G 2 in K n , then we say there is a packing of G 1 and G 2 .  ...  Their conjecture can be restated in the following complementary form: Conjecture 1 (Bollob as-Eldridge) If G is a graph on n vertices with (G) kn ?  ... 
doi:10.1007/s00493-003-0013-4 fatcat:s4lwmzrw5fa5ba2uyf5fhjhbyy

On Cyclic Packing of a Tree

Stephan Brandt, Mariusz Woźniak
2004 Graphs and Combinatorics  
We prove that there exists a packing of bn=2c copies of a tree of size dn=2e into K n . Moreover, the proof provides an easy algorithm.  ...  Conjectures and Results The main motivation of the paper is the following well-known conjecture of Bolloba´s and Eldridge ([1]). Conjecture 1.  ...  Observe that in this case the total number of edges we pack into K n is maximum (with respect to the Conjecture of Boloba´s and Eldridge).  ... 
doi:10.1007/s00373-004-0583-y fatcat:gtfakzbqgbcuzfzswp255uirfy

Béla Bollobás, Extremal Graph Theory (Academic Press, 1978), 488 pp., £19·50

I. Anderson
1980 Proceedings of the Edinburgh Mathematical Society  
The book is in three sections. The first two concern, respectively, geometrical and combinatorial problems for a single lattice. The third concerns problems on sets of lattices.  ...  He gives references to the literature for all proofs. As a result, the reader who wishes to get the full flavour of the subject will need access to a well-stocked library.  ...  The chapter then continues with/-factors, and recent work on matchings in graphs with given maximum and minimum degrees.  ... 
doi:10.1017/s0013091500003709 fatcat:4lf44qzutnce7gtrtwscojr35y

Paul Catlin 1948–1995

Arthur M. Hobbs, Hong-Jian Lai, Neil Robertson
2001 Discrete Mathematics  
In 1976, Catlin ([C0,C8]), and independently Bollobà as and Eldridge [BoEL] , conjectured that there is a packing of G 1 and G 2 if ( (G 1 ) + 1)( (G 2 ) + 1)6n + 1.  ...  For a matching M 3 consisting of three edges, (M 3 ) denotes the sum of the degrees of the six vertices incident with M 3 .  ... 
doi:10.1016/s0012-365x(00)00065-0 fatcat:ubsvt26bxfhq3nvdobmhgzldoe

The Bollobás-Eldridge-Catlin conjecture for even girth at least 10 [article]

Wouter Cames van Batenburg, Ross J. Kang
2017 arXiv   pre-print
A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that, if (Δ(G_1)+1) (Δ(G_2)+1) < n+1, then G_1 and G_2 pack.  ...  Two graphs G_1 and G_2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint.  ...  Conjecture 1.1 (Bollobás and Eldridge [2] and Catlin [7] ) If G 1 and G 2 are graphs on n vertices with respective maximum degrees ∆ 1 and ∆ 2 such that (∆ 1 + 1)(∆ 2 + 1) ≤ n + 1, then they pack.  ... 
arXiv:1703.05149v1 fatcat:cm7lo76hovbfloxw7ekmqahbqa

Page 2695 of Mathematical Reviews Vol. , Issue 83h [page]

1983 Mathematical Reviews  
Bollobas, Eldridge and the author have conjectured that two graphs are mutually placeable if the sum of the product and the sum of their maximum valences is at most p.  ...  Upper and lower bounds are obtained for the number of triangles in a 2-connected or 3-connected planar graph in terms of its face degrees.  ... 

Packing graphs of bounded codegree [article]

Wouter Cames van Batenburg, Ross J. Kang
2016 arXiv   pre-print
A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that, if (Δ_1(G)+1) (Δ_2(G)+1) < n+1, then G_1 and G_2 pack.  ...  In particular, we prove for all t > 2 that, if G_1 contains no copy of the complete bipartite graph K_2,t and Δ_1 > 17 t ·Δ_2, then (Δ_1(G)+1) (Δ_2(G)+1) < n+1 implies that G_1 and G_2 pack.  ...  The maximum codegree ∆ ∧ (G) of a graph G is the maximum over all vertex pairs of their common degree, i.e. ∆ ∧ (G) < t if and only if G contains no copy of the complete bipartite graph K 2,t .  ... 
arXiv:1605.05599v1 fatcat:ikdcqaynf5fj5oyye5owmiqbwi

Packing Graphs of Bounded Codegree

WOUTER CAMES VAN BATENBURG, ROSS J. KANG
2018 Combinatorics, probability & computing  
A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G 1) + 1)(Δ(G 2) + 1) ⩽ n + 1, then G 1 and G 2 pack.  ...  In particular, we prove for all t ⩾ 2 that if G 1 contains no copy of the complete bipartite graph K 2,t and Δ(G 1) &gt; 17t · Δ(G 2), then (Δ(G 1) + 1)(Δ(G 2) + 1) ⩽ n + 1 implies that G 1 and G 2 pack  ...  Acknowledgement We are grateful to the anonymous referee for their careful reading and helpful comments.  ... 
doi:10.1017/s0963548318000032 fatcat:mxwqt2awkrecznxcgwom4w3g7m

On the Bollobás–Eldridge Conjecture for Bipartite Graphs

B. CSABA
2007 Combinatorics, probability & computing  
Let G be a simple graph on n vertices. A conjecture of Bollobás and Eldridge [5] asserts that if δ(G) ≥ kn−1 k+1 then G contains any n vertex graph H with ∆(H) = k.  ...  This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-up Lemma.  ...  Acknowledgment The author would like to thank János Komlós for suggesting the problem, and for many helpful conversations, and the anonymous referees for improving the quality of the presentation.  ... 
doi:10.1017/s0963548307008395 fatcat:ipibgishpzeqzg5ddrz32zvxja

PSP volume 79 issue 3 Cover and Back matter

1976 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
As one concerned with mathematics, a membership mathematics to consider in SIAM offers you a broad publications program with a healthy balance of emphasis between abstract and applied mathematics.  ...  invites those interested in the application of membership.  ...  E. and Bollobas, B. Maximal matchings in graphs with given minimum and maximum degrees 221 Elliott, P. D. T. A.  ... 
doi:10.1017/s0305004100052385 fatcat:24iyis2f3ncylnwjoriejeoohm

Embedding graphs into larger graphs: results, methods, and problems [article]

Miklós Simonovits, Endre Szemerédi
2019 arXiv   pre-print
Extremal Graph Theory is a very deep and wide area of modern combinatorics.  ...  --- for some other reasons --- are very close to us.  ...  We thank above all to József Balogh and András Gyárfás, and also to Zoltán Füredi, János Pach, Jan Hladký, Zoltán L.  ... 
arXiv:1912.02068v1 fatcat:2rgpm6wbmvafta74cmcbbxhcs4

2-factors in dense graphs

Noga Alon, Eldar Fischer
1996 Discrete Mathematics  
A conjecture of Sauer and Spencer states that any graph G on n vertices with minimum degree at least~n contains any graph H on n vertices with maximum degree 2 or less.  ...  This conjecture is proven here for all sufficiently large n.  ...  Zwick and an anonymous referee for helpful comments.  ... 
doi:10.1016/0012-365x(95)00242-o fatcat:irtarvsvqjadxfzn5js2mkk3a4

Packings of graphs and applications to computational complexity

B Bollobás, S.E Eldridge
1978 Journal of combinatorial theory. Series B (Print)  
Let G1 and GS be graphs with n vertices. If there are edge-disjoint copies of G1 and G, with the same n vertices, then we say there is a packing of G, and Ge .  ...  In particular, we show that, for subgraphs of tournaments, the property of containing a sink is a monotone property with minimal computational complexity. 105  ...  ACKNOWLEDGMENT We would like to express our gratitude to the referee for his excellent work, and to Mr. Paul Catlin for pointing out an error in an earlier version of the first conjecture in 52.  ... 
doi:10.1016/0095-8956(78)90030-8 fatcat:tya3x7tz5fcalbdayubtjvj3la

The Turán number of sparse spanning graphs [article]

Noga Alon, Raphael Yuster
2014 arXiv   pre-print
This generalizes a classical result of Ore for the case H=C_n, and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs.  ...  For a graph H, the extremal number ex(n,H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H.  ...  Acknowledgment We would like to thank Roman Glebov for helpful comments. We also thank Andrew McConvey for pointing out an inaccurate point in an earlier published version of the paper.  ... 
arXiv:1404.1182v1 fatcat:ylo4oy37pfbxdi3wh5vk4mpgo4

The Turán number of sparse spanning graphs

Noga Alon, Raphael Yuster
2013 Journal of combinatorial theory. Series B (Print)  
This generalizes a classical result of Ore for the case H = C n , and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs.  ...  For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H.  ...  Acknowledgment We would like to thank Roman Glebov for helpful comments.  ... 
doi:10.1016/j.jctb.2013.02.002 fatcat:ctryleuinbfznebyjx6bb2qiuy
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