Rainbow triangles in arc-colored tournaments

Abstract

Let T n be an arc-colored tournament of order n. The maximum monochromatic indegree ∆ −mon (T n ) (resp. outdegree ∆ +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 tournamentWe 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.

Rainbow triangles in arc-colored digraphs

“…Li et al [15] further proved that if G is an edge-colored graph of order n satisfying d c (u) + d c (v) ≥ n + 1 for every edge uv ∈ E(G), then it contains a rainbow triangle. In [16], Li et al gave some maximum monochromatic degree conditions for an arc-colored strongly connected tournament T n to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. For more results on rainbow cycles, see [1,4,5,10].…”