On the vertices of a 3-partite tournament not in triangles

Abstract

Let T be a 3-partite tournament and F3(T ) be the set of vertices of T not in triangles. We prove that, if the global irregularity of T , ig (T ), is one and |F3(T )| > 3, then F3(T ) must be contained in one of the partite sets of T and |F3(T )| ≤  k+1 4  + 1, which implies |F3(T )| ≤  n+5 12  + 1, where k is the size of the largest partite set and n the number of vertices of T . Moreover, we give some upper bounds on the number, as well as results on the structure of said vertices within the digraph, depending on its global irregularity.