Tournaments are orientations of the complete graph. The directed Ramsey number R(k) is minimum vertices a tournament must have to be guaranteed contain transitive subtournament size k, which we denote by $$ TT _k$$ . We include computer-assisted proof conjecture Sanchez-Flores in Graphs Combinatorics 14(2), 181–200 (1998), that all _6$$ -free tournaments on 24 and 25 subtournaments ST _{27}$$ ,...

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This...

The H-free process, for some fixed graph H , is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c > 0, with high pro...

Abstract The size-Ramsey number of a graph F is the smallest edges in G with Ramsey property for , that is, any 2-colouring contains monochromatic copy . We prove grid on n × vertices bounded from above by 3+o(1)

Given a graph G and positive integer k, define the Gallai—Ramsey number to be minimum of vertices n such that any k-edge coloring Kn contains either rainbow (all different colored) triangle or monochromatic copy G. In this paper, we obtain exact values numbers for union two stars in many cases bounds other cases. This work represents first class disconnected graphs considered as desired subgraph.