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Graph classes with linear Ramsey numbers
[article]

2020
*
arXiv
*
pre-print

We also apply the notion of linearity to

arXiv:1910.12109v3
fatcat:hwcd736dqrdktc5p6qx3hajote
*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. ...##
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On number of different sized induced subgraphs of Bipartite-Ramsey graphs
[article]

2021
*
arXiv
*
pre-print

In this paper, we investigate the set of sizes of induced

arXiv:2109.08485v1
fatcat:upnwgt3qjrgxvfqlklct3fsugi
*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. ...##
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Degree bipartite Ramsey numbers
[article]

2019
*
arXiv
*
pre-print

In this note, we show that r_Δ(K_m,n;s) is linear

arXiv:1909.00147v1
fatcat:ixt3uiuysbdvxbi75uuayyxgti
*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. ...##
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Generalized local colorings of graphs

1992
*
Journal of combinatorial theory. Series B (Print)
*

We prove the

doi:10.1016/0095-8956(92)90049-4
fatcat:scavypuyd5dppafxf6hxjpwiyy
*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. ...##
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Local k-colorings of graphs and hypergraphs

1987
*
Journal of combinatorial theory. Series B (Print)
*

The size

doi:10.1016/0095-8956(87)90017-7
fatcat:nep3mrrrybehdjmxahrkxdr5vm
*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(; ). ...##
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On the Ramsey Numbers for Bipartite Multigraphs
[article]

2003
*
arXiv
*
pre-print

(In words, the induced

arXiv:cs/0305006v1
fatcat:olkho6wkwrcy5p6a6l2n32ee6q
*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. ...##
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Star-path and star-stripe bipartite Ramsey numbers in multicoloring

2015
*
Transactions on Combinatorics
*

,H t , then at least

doaj:c8d06d8051044cb2af9bd1950a7fc4ed
fatcat:ue26pam6kzaupiobpswcpmgsze
*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*. ...##
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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*. ...##
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On globally sparse Ramsey graphs

2013
*
Discrete Mathematics
*

Rödl and Ruciński asked how globally sparse (F,r)-

doi:10.1016/j.disc.2013.07.023
fatcat:pcmxavyinng75oza2urf72v2pi
*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 ...##
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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*...##
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Star-path bipartite Ramsey numbers

1998
*
Discrete Mathematics
*

For

doi:10.1016/s0012-365x(97)00205-7
fatcat:od2d7sd4djd5tcy6kppbxjiyta
*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 ..... ...##
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Chromatic Number is Ramsey Distinguishing
[article]

2019
*
arXiv
*
pre-print

Two

arXiv:1909.02590v1
fatcat:gpzfz3b3dvcftb67mukk4tfi7u
*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 ...##
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Bipartite Ramsey numbers and Zarankiewicz numbers

2000
*
Discrete Mathematics
*

The

doi:10.1016/s0012-365x(99)00370-2
fatcat:ackm7frfofdcle5gqxbiokzonm
*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*. ...##
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On partitions of hereditary properties of graphs

2006
*
Discussiones Mathematicae Graph Theory
*

In this paper a concept Q-

doi:10.7151/dmgt.1330
fatcat:s6ulq5j6fbamrimlawhbqz6m4u
*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. ...##
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A new upper bound for the bipartite Ramsey problem

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

doi:10.1002/jgt.20317
fatcat:hkhzayjcwvevjnqhmnwfpldyqq
*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 . ...
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