On n-stars in colorings and orientations of graphs

An n-star S in a graph G is the union of geodesic intervals I1,…,Ik with common end O such that the subgraphs I1∖{O},…,Ik∖{O} are pairwise disjoint and l(I1)+…+l(Ik)=n. If the edges of G are oriented, S is directed if each ray Ii is directed. For natural number n,r, we construct a graph G of diam(G)=n such that, for any r-coloring and orientation of E(G), there exists a directed n-star with monochrome rays of pairwise distinct colors

Ramsey-Type Results for Oriented Trees

. For a graph G and a digraph ~ H, we write G! ~ H (respectively, G a ! ~ H) if every orientation (respectively, acyclic orientation) of the edges of G results in an induced copy of ~ H. In this note we study how small the graphs G such that G! ~ H or such that G a ! ~ H may be, if ~ H is a given oriented tree ~ T n on n vertices. We show that there is a graph on O(n 4 log n) vertices and O(n 6 (log n) 2 ) edges such that G! ~ T n for every n-vertex oriented tree ~ T n . We also prove that there exists a graph G with O(n 2 log n) vertices and O(n 3 (log n) 2 ) edges such that G a ! ~ T n for any such tree ~ T n . This last result turns out to be nearly best possible as it is shown that any graph G with G a ! ~ P n , where ~ P n is the directed path of order n, has more than n 2 =2 vertices and more than bn=3c 3 edges if n 3. 1991 Mathematics subject classications: Primary 05C55, 05C20, 05C05; Secondary 05C80 Key words: Ramsey theor..