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Graph classes with linear Ramsey numbers [article]

Bogdan Alecu, Aistis Atminas, Vadim Lozin, Viktor Zamaraev
2020 arXiv   pre-print
We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.  ...  The Ramsey number R_X(p,q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q.  ...  Then X contains all bipartite graphs without cycles of length at most k, and hence bipartite Ramsey numbers are not linear in X by Corollary 1.  ... 
arXiv:1910.12109v3 fatcat:hwcd736dqrdktc5p6qx3hajote

On number of different sized induced subgraphs of Bipartite-Ramsey graphs [article]

Mantas Baksys, Xuanang Chen
2021 arXiv   pre-print
In this paper, we investigate the set of sizes of induced subgraphs of bipartite graphs.  ...  We introduce the definition of C-Bipartite-Ramsey graphs, which is closely related to Ramsey graphs and prove that in 'most' cases, these graphs have multiplication tables of Ω(e(G)) in size.  ...  graph, which states that a simple graph G on n vertices is C-Ramsey if it does not contain K m or K m as an induced subgraph for m ≥ C log n.  ... 
arXiv:2109.08485v1 fatcat:upnwgt3qjrgxvfqlklct3fsugi

Degree bipartite Ramsey numbers [article]

Ye Wang, Yusheng Li, Yan Li
2019 arXiv   pre-print
In this note, we show that r_Δ(K_m,n;s) is linear on n with m fixed. We also determine br_Δ(G;s) where G are trees, including stars and paths, and complete bipartite graphs.  ...  The degree Ramsey number r_Δ(G;s) is defined to be min{Δ(H):Hs G}, and the degree bipartite Ramsey number br_Δ(G;s) is defined to be min{Δ(H):Hs G χ(H)=2}.  ...  Let us consider a complete bipartite graph K M,N on bipartition (A, B) and an edge partition (E 1 , E 2 , . . . , E s ). Without loss of generality, assume that |E 1 | ≥ M N/s.  ... 
arXiv:1909.00147v1 fatcat:ixt3uiuysbdvxbi75uuayyxgti

Generalized local colorings of graphs

Miroslaw Truszczyński
1992 Journal of combinatorial theory. Series B (Print)  
We prove the Ramsey property for such colorings, establish conditions for the density property and the bipartite version of the Ramsey theorem to hold, and prove the induced variant of the Ramsey theorem  ...  A local (H, k)-coloring of a graph G is a coloring of the edges of G such that edges of no subgraph of G isomorphic to a subgraph of H are colored with more than k colors.  ...  For every bipartite graph H and every k, there exists a bipartite graph G such that when G is (locally) k-colored, then it contains a monochromatic copy of H as an induced subgraph of G.  ... 
doi:10.1016/0095-8956(92)90049-4 fatcat:scavypuyd5dppafxf6hxjpwiyy

Local k-colorings of graphs and hypergraphs

A Gyárfás, J Lehel, J Nešetřil, V Rödl, R.H Schelp, Zs Tuza
1987 Journal of combinatorial theory. Series B (Print)  
The size Ramsey number F'(G) of a graph G was introduced in [8] . Using the notion of minimal Ramsey graphs (see Section 5 ); ?(G) = min{ IE(H)J: HE&(G)}.  ...  Beck for the discussion on this remark.  ...  It is easy to see that B(p,) is a universal bipartite 582b/43/2-2 GYikRFriSET AL. graph in the sense that all bipartite graphs appear as an induced subgraph of some B(; ).  ... 
doi:10.1016/0095-8956(87)90017-7 fatcat:nep3mrrrybehdjmxahrkxdr5vm

On the Ramsey Numbers for Bipartite Multigraphs [article]

Ming-Yang Chen, Hsueh-I. Lu, Hsu-Chun Yen
2003 arXiv   pre-print
(In words, the induced subgraph with respect to color c is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs.  ...  Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., m) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring  ...  Acknowledgments We thank Gerard Jennhwa Chang for very helpful comments on a preliminary version of this paper.  ... 
arXiv:cs/0305006v1 fatcat:olkho6wkwrcy5p6a6l2n32ee6q

Star-path and star-stripe bipartite Ramsey numbers in multicoloring

Ghaffar Raeisi
2015 Transactions on Combinatorics  
,‎H t ‎, ‎then at least one H i has a subgraph isomorphic to G i ‎. ‎In this paper‎, ‎we study the multicolor bipartite Ramsey number bR(G 1 ‎,‎G 2 ,...‎,‎G t ) ‎, ‎in the case that G 1 ‎,‎G 2 ,...‎  ...  ‎For given bipartite graphs G 1 ‎,‎G 2 ,...‎,‎G t , the bipartite Ramsey number bR(G 1 ‎,‎G 2 ,...‎  ...  Bipartite Ramsey problems deal with the same questions but the graph explored is the complete bipartite graph instead of the complete graph. Let G 1 , G 2 , . . . , G t be bipartite graphs.  ... 
doaj:c8d06d8051044cb2af9bd1950a7fc4ed fatcat:ue26pam6kzaupiobpswcpmgsze

Page 4602 of Mathematical Reviews Vol. , Issue 82k [page]

1982 Mathematical Reviews  
Now let B, and B, be two bipartite graphs; the bipartite Ramsey set B(B,,B,) is defined as the set of pairs (m,n), m<n, such that Kn > (B,, Bz) and neither Kot. nor Kant have this property.  ...  COMBINATORICS 4602 In other words, every graph G with at least one edge can be uniquely reconstructed from its maximal subgraphs.  ... 

On globally sparse Ramsey graphs

Torsten Mütze, Ueli Peter
2013 Discrete Mathematics  
Rödl and Ruciński asked how globally sparse (F,r)-Ramsey graphs G can possibly be, where the density of G is measured by the subgraph H⊆ G with the highest average degree.  ...  In this work we determine the Ramsey density up to some small error terms for several cases when F is a complete bipartite graph, a cycle or a path, and r≥ 2 colors are available.  ...  In this section we will construct a sparse (P , r)-Ramsey graph by using the bipartite graph G = G(n, k, m) = (N We call a coloring of the edges of a complete bipartite graph with vertex partition A and  ... 
doi:10.1016/j.disc.2013.07.023 fatcat:pcmxavyinng75oza2urf72v2pi

Page 49 of Mathematical Reviews Vol. , Issue 80A [page]

1980 Mathematical Reviews  
This is related to some Ramsey-type problems for bipartite graphs.  ...  For two graphs A and B, the connected Ramsey number r,(A, B) is the least integer n (n > 4) such that, for every coconnected graph G on at least n vertices, either A is a subgraph of G or B is a subgraph  ... 

Star-path bipartite Ramsey numbers

Johannes H. Hattingh, Michael A. Henning
1998 Discrete Mathematics  
For bipartite graphs G1,G2 ..... Gk, the bipartite Ramsey number b(GI,G2,...  ...  In this note, we establish the exact value of the bipartite Ramsey number b (Pm,Kl") for all integers re, n>.2, where Pm denotes a path on m vertices. (~)  ...  If G is a subgraph of H, then the graph H -E(G) is the complement of G relative to H. For bipartite graphs G1,G2 ..... Gk, the (generalized) bipartite Ramsey number b(G1,G2 .....  ... 
doi:10.1016/s0012-365x(97)00205-7 fatcat:od2d7sd4djd5tcy6kppbxjiyta

Chromatic Number is Ramsey Distinguishing [article]

Michael Savery
2019 arXiv   pre-print
Two graphs H_1 and H_2 are Ramsey equivalent if any graph G is Ramsey for H_1 if and only if it is Ramsey for H_2.  ...  A graph G is Ramsey for a graph H if every colouring of the edges of G in two colours contains a monochromatic copy of H.  ...  Indeed, the chromatic Ramsey number of K 4 , the complete graph on four vertices, is 18 (since we can 2-edge-colour any 17vertex-colourable graph without a monochromatic K 4 by naturally extending from  ... 
arXiv:1909.02590v1 fatcat:gpzfz3b3dvcftb67mukk4tfi7u

Bipartite Ramsey numbers and Zarankiewicz numbers

Wayne Goddard, Michael A. Henning, Ortrud R. Oellermann
2000 Discrete Mathematics  
The bipartite Ramsey number b(m, n) is the least positive integer b such that if the edges of K(b, b) are coloured with red and blue, then there always exists a blue K(m, m) or a red K(n, n).  ...  The Zarankiewicz number z(s, m) is the maximum number of edges in a subgraph of K(s, s) that does not contain K(m, m) as a subgraph.  ...  Introduction Zarankiewicz numbers [15] involve bounds on the maximum number of edges in a bipartite graph without a particular subgraph.  ... 
doi:10.1016/s0012-365x(99)00370-2 fatcat:ackm7frfofdcle5gqxbiokzonm

On partitions of hereditary properties of graphs

Mieczysław Borowiecki, Anna Fiedorowicz
2006 Discussiones Mathematicae Graph Theory  
In this paper a concept Q-Ramsey Class of graphs is introduced, where Q is a class of bipartite graphs. It is a generalization of wellknown concept of Ramsey Class of graphs.  ...  Some Q-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that T 2 , the class of all outerplanar graphs, is not D 1 -Ramsey Class (Theorem 3).  ...  Let F be a connected bipartite graph and Q =Forb(F ). Then O k is a Q-Ramsey Class for k ≥ 2. P roof.  ... 
doi:10.7151/dmgt.1330 fatcat:s6ulq5j6fbamrimlawhbqz6m4u

A new upper bound for the bipartite Ramsey problem

David Conlon
2008 Journal of Graph Theory  
We consider the following question: how large does n have to be to guarantee that in any two-colouring of the edges of the complete graph K n,n there is a monochromatic K k,k ?  ...  Consider the induced red bipartite subgraph on the vertex sets M and N .  ...  Recall that the Zarankiewicz number z(n, n, k, l) is the maximum number of edges that one can have in a bipartite graph, both of whose vertex sets are of size n, without containing a K k,l .  ... 
doi:10.1002/jgt.20317 fatcat:hkhzayjcwvevjnqhmnwfpldyqq
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