Rainbow Triangles in Arc-Colored Tournaments

Abstract

Let \(T_{n}\) be an arc-colored tournament of order n. The maximum monochromatic indegree \(\varDelta ^{-mon}(T_{n})\) (resp. outdegree \(\varDelta ^{+mon}(T_{n})\)) 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 \(\varDelta ^{-mon}(T_n)\le \varDelta ^{+mon}(T_n)\) is contained in at least \(\frac{\delta (v)(n-\delta (v)-i(T_n))}{2}-[\varDelta ^{-mon}(T_{n})(n-1)+\varDelta ^{+mon}(T_{n})d^+(v)]\) rainbow triangles, where \(\delta (v)=\min \{d^+(v), d^-(v)\}\). 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.