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Page 650 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
In this paper the au- thors find exact values of r(7T,,:,S,41) where T,,; and S,,; are trees with maximum degree p — | and q — 1, respectively (note that K\,, is a tree with maximum degree p).  ...  This paper con- tains two main results dealing with graph Ramsey numbers for trees. The first improves a result of S. A.  ... 

Some Ramsey–Turán type problems and related questions

R.H. Schelp
2012 Discrete Mathematics  
Specific results, conjectures, and questions with suggested values for c are considered when G is an odd cycle, path, or tree of limited maximum degree.  ...  For which graphs G does there exist a constant 0 < c < 1 such that when H is a graph of order the Ramsey number r(G) with δ(H) > c|H|, then any 2-edge coloring of H contains a monochromatic copy of G?  ...  Ramsey number for any tree on n vertices when l = 2k = 2n/3.  ... 
doi:10.1016/j.disc.2011.09.015 fatcat:gjjizn7b6zaqzjeypu5plj5tom

Tree containment and degree conditions [article]

Maya Stein
2020 arXiv   pre-print
We survey results and open problems relating degree conditions with tree containment in graphs, random graphs, digraphs and hypergraphs, and their applications in Ramsey theory.  ...  As noted above, the Ramsey number for paths differs significantly from the Ramsey number for stars with the same number of edges.  ...  Let us now briefly look at results and questions for multi-colour Ramsey numbers of trees.  ... 
arXiv:1912.04004v2 fatcat:tgw5raavandr3bkfamqlt7ers4

Page 2010 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
H. (1-MEMP) Small order graph-tree Ramsey numbers. Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986). Discrete Math. 72 (1988), no. 1-3, 119-127.  ...  The au- thors prove that (3) there exists a positive constant c such that for every tree T of order n and maximum degree A(T) = m, r(K3.3, T) < max{n + [cn'/*), r(K3,3, Ki.m)}.  ... 

Page 2345 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
W. (3-WTRL-B; Waterloo, ON) Ramsey numbers for trees of small maximum degree. (English summary) Special issue: Paul Erdés and his mathematics. Combinatorica 22 (2002), no. 2, 287-320.  ...  Summary: “Given a positive integer n and a family ¥ of graphs, Graph theory 20034:05144 the anti-Ramsey number f(n,¥) is the maximum number of colors in an edge-coloring of K, such that no subgraph of  ... 

Explicit Construction of Linear Sized Tolerant Networks [chapter]

N. Alon, F.R.K. Chung
1988 Annals of Discrete Mathematics  
For every > 0 and every integer m > 0, we construct explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion of their vertices or edges, the remaining  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  of its edges contains every tree of size m and maximum degree k.  ... 
doi:10.1016/s0167-5060(08)70766-0 fatcat:kof5ehgpobe65lthp6v3yffqtq

Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
1988 Discrete Mathematics  
For every > 0 and every integer m > 0, we construct explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion of their vertices or edges, the remaining  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  of its edges contains every tree of size m and maximum degree k.  ... 
doi:10.1016/0012-365x(88)90189-6 fatcat:2h6m2f44uzac5fn23qrwi3x4hy

Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
2006 Discrete Mathematics  
For every > 0 and every integer m > 0, we construct explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2 ), such that after removing any (1 − ) portion of their vertices or edges, the remaining  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  of its edges contains every tree of size m and maximum degree k.  ... 
doi:10.1016/j.disc.2006.03.025 fatcat:sbp4m3smcbccxkrvvyplfuhfla

Some Complete Bipartite Graph—Tree Ramsey Numbers [chapter]

S. Burr, P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp
1988 Annals of Discrete Mathematics  
For a = 2, this Ramsey number is completely determined by r(IC ~J,K~,~J where m = A(T).  ...  Dedicated to the memory of G. A. Diruc We investigate r(K"" T) for a = 2 and a = 3, where T is an arbitrary tree of order n.  ...  . , zk-r have degree two in G. C&tree Ramsey numbers Let T be a tree of order n and maximum degree A(T) = m.  ... 
doi:10.1016/s0167-5060(08)70452-7 fatcat:c6tiyslp4fgvlom5fupygcpx7e

On-line Ramsey Theory for Bounded Degree Graphs

Jane Butterfield, Tracy Grauman, William B. Kinnersley, Kevin G. Milans, Christopher Stocker, Douglas B. West
2011 Electronic Journal of Combinatorics  
The on-line degree Ramsey number $\mathring {R}_\Delta(G)$ is the least $k$ such that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$.  ...  a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring  ...  For this last case, inscribing an r-cycle yields a desired cycle, 8-weighted.  ... 
doi:10.37236/623 fatcat:rx6c6374j5h4rgwhtnaqy6pa6m

Degree Ramsey numbers for cycles and blowups of trees

Tao Jiang, Kevin G. Milans, Douglas B. West
2013 European journal of combinatorics (Print)  
We prove that the 2-color degree Ramsey number is at most 96 for every even cycle and at most 3458 for every odd cycle.  ...  The notion has been studied with ρ(G) being the clique number [15, 24] , the chromatic number [6, 30, 31] , the number of edges (called the size Ramsey number) [2, 3, 9, 11, 16, 26] , and the maximum degree  ...  By Corollary 1.4, within the family of trees, the degree Ramsey number is bounded by a function of the number of colors and the maximum degree. Section 5 extends this statement to larger families.  ... 
doi:10.1016/j.ejc.2012.08.004 fatcat:zlrsa2yhjvgs3lxl3wauhd7hoi

The vertex size-Ramsey number

Andrzej Dudek, Linda Lesniak
2016 Discrete Mathematics  
Finally, we prove thatR v (T, r) = O r (∆ 2 n) for any tree of order n and maximum degree ∆, which is only off by a factor of ∆ from the best possible.  ...  In this paper, we study an analogue of size-Ramsey numbers for vertex colorings.  ...  Acknowledgement We are grateful to David Conlon and Jacob Fox for fruitful discussion, and in particular for pointing out Theorem 1.11.  ... 
doi:10.1016/j.disc.2016.02.001 fatcat:26fay3jqkzhnrh4hekfztqcxaa

On-line Ramsey numbers for paths and stars

J. A. Grytczuk, H. A. Kierstead, P. Prałat
2008 Discrete Mathematics & Theoretical Computer Science  
The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H.  ...  Graphs and Algorithms International audience We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by  ...  The size-Ramsey number of H can be bounded from below in terms of τ (H) and the maximum degree ∆(H) of a graph H. A similar bound holds in the on-line case.  ... 
doi:10.46298/dmtcs.427 fatcat:4k6pxnw4hveklfywwrctmvygvq

Page 721 of Mathematical Reviews Vol. 54, Issue 3 [page]

1977 Mathematical Reviews  
The author gives a necessary and sufficient condition for the existence of Ramsey numbers, and, when they exist, an upper bound function. Ramsey numbers for some classes of digraphs are found.  ...  Some Ramsey numbers for directed graphs. Discrete Math. 9 (1974), 313-321.  ... 

On Some Three-Color Ramsey Numbers

Wai Chee Shiu, Peter Che Bor Lam, Yusheng Li
2003 Graphs and Combinatorics  
In this paper we study three-color Ramsey numbers. Let K i,j denote a complete i by j bipartite graph.  ...  We shall show that (i) for any connected graphs G 1 , G 2 and G 3 , if is the chromatic surplus of G 3 ; (ii)(k + m − 2)(n − 1) + 1 ≤ r(K 1,k , K 1,m , K n ) ≤ (k + m − 1)(n − 1) + 1, and if k or m is  ...  Acknowledgment: The authors are grateful to the referee for his valuable comments.  ... 
doi:10.1007/s00373-002-0495-7 fatcat:5tkeenaawzc6pmp5fuep2m4d2q
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