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Subdivided graphs have linear Ramsey numbers Bellcore, Morristown, NJ 07960, USA
 

Summary: Subdivided graphs have linear Ramsey numbers
Noga Alon
Bellcore, Morristown, NJ 07960, USA
and Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
It is shown that the Ramsey number of any graph with n vertices in which no two vertices
of degree at least 3 are adjacent is at most 12n. In particular, the above estimate holds for the
Ramsey number of any n-vertex subdivision of an arbitrary graph, provided each edge of the
original graph is subdivided at least once. This settles a problem of Burr and Erd¨os.
1 Introduction
The Ramsey number of a graph G, denoted by r(G), is the minimum integer t such that in any
coloring of the edges of the complete graph Kt on t vertices by red and blue, there is always a
monochromatic copy of G. We say that a family of graphs F is a linear family if there is a constant
c > 0 such that for every member G of F, r(G) cn, where n is the number of vertices of G. In
this note we prove the following result, conjectured by Burr and Erd¨os ([1], page 236) in 1973.
Theorem 1.1 The family of all graphs that have no two adjacent vertices of degree at least 3 is a
linear family.
This strengthens a result of [1] that asserts that the family of all graphs in which the distance

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics