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

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 r̂(F1,F2) for many pairs (F1,F2) 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.