Ramsey-type Numbers for Degree Sequences
نویسندگان
چکیده
A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G1 and G2, we consider the smallest integer n such that for every n-term graphic sequence π, there is some graph G with degree sequence π with G1 ⊆ G or with G2 ⊆ G. When the phrase “some graph” in the prior sentence is replaced with “all graphs” the smallest such integer n is the classical Ramsey number r(G1, G2) and thus we call this parameter for degree sequences the potential-Ramsey number and denote it rpot(G1, G2). In this paper, we give exact values for rpot(Kn,Kt), rpot(Cn,Kt), and rpot(Pn,Kt) and consider situations where rpot(G1, G2) = r(G1, G2).
منابع مشابه
Zarankiewicz Numbers and Bipartite Ramsey Numbers
The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...
متن کاملThe Ramsey numbers of large trees versus wheels
For two given graphs G1 and G2, the Ramseynumber R(G1,G2) is the smallest integer n such that for anygraph G of order n, either $G$ contains G1 or the complementof G contains G2. Let Tn denote a tree of order n andWm a wheel of order m+1. To the best of our knowledge, only R(Tn,Wm) with small wheels are known.In this paper, we show that R(Tn,Wm)=3n-2 for odd m with n>756m^{10}.
متن کاملRamsey numbers for bipartite graphs with small bandwidth
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assump...
متن کامل3-Uniform hypergraphs of bounded degree have linear Ramsey numbers
Chvátal, Rödl, Szemerédi and Trotter [1] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform ‘pseudo-random’ hypergraphs. keywords: hypergraphs; regularity lemm...
متن کاملOn the Size-Ramsey Number of Hypergraphs
The size-Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2-edge-coloring of H yields a monochromatic copy of G. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for k-uniform hypergraphs. Analogous to the graph case, we cons...
متن کامل