Rainbow Triangles in Arc-Colored Tournaments

Let T_n be an arc-colored tournament of order n. The maximum monochromatic indegree Δ^-mon(T_n) (resp. outdegree Δ^+mon(T_n)) of T_n is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of T_n. The irregularity i(T_n) of T_n is the maximum difference between the indegree and outdegree of a vertex of T_n. A subdigraph H of an arc-colored digraph D is called rainbow if each pair of arcs in H have distinct colors. In this paper, we show that each vertex v in an arc-colored tournament T_n with Δ^-mon(T_n) ≤ Δ^+mon(T_n) is contained in at least δ(v)(n-δ(v)-i(Tn))2-[Δ-mon(Tn)(n-1)+Δ+mon(Tn)d+(v)] rainbow triangles, where δ(v) = min { d⁺(v) , d^-(v) }. We also give some maximum monochromatic degree conditions for T_n to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible.