Two-edge colorings of graphs with bounded degree in both colors

Abstract

Let F be the family of all graphs of maximum degree k+ℓ which can be red–blue edge colored with each of its vertices incident to at most k red edges and at most ℓ blue edges. Let m( k,ℓ) be the maximum number such that every graph with at most m( k,ℓ) vertices of maximum degree k+ℓ is in F. This paper determines m( k,ℓ) except when one of k and ℓ is odd and the other even, in which case best known bounds are given. These values of m( k,ℓ) are used to determine the size Ramsey number r ̂ (F 1,F 2) for many pairs ( F 1, F 2) of star forests, which gives a partial solution to a conjecture of Burr et al. (Proceedings of the Koninklijke Nederlandse Academie van Wetenschappen, Series A, Vol. 81(2), 1978, p. 187.) made in 1978.