Ramsey numbers for local colorings

Abstract

The concept of a localk-coloring of a graphG is introduced and the corresponding localk-Ramsey numberr loc k (G) is considered. A localk-coloring ofG is a coloring of its edges in such a way that the edges incident to any vertex ofG are colored with at mostk colors. The numberr loc k (G) is the minimumm for whichK m contains a monochromatic copy ofG for every localk-coloring ofK m . The numberr loc k (G) is a natural generalization of the usual Ramsey numberr k (G) defined for usualk-colorings. The results reflect the relationship betweenr k (G) andr loc k (G) for certain classes of graphs.