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Rainbow Triangles in Arc-Colored Tournaments
Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2021-05-14 , DOI: 10.1007/s00373-021-02313-w
Wei Li , Shenggui Zhang , Yandong Bai , Ruonan Li

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.



中文翻译:

彩虹色比赛中的彩虹三角形

\(T_ {n} \)n阶的弧形锦标赛。最大单色入度\(\ varDelta ^ { -周一}(T_ {N})\) (分别出度。\(\ varDelta ^ {+周一}(T_ {N})\) )的\(T_ {N} \)是入射到\(T_ {n} \)顶点的相同颜色的弧内最大弧数(分别为弧外。不规则\(I(T_ {N})\)\(T_ {N} \)是一个顶点的入度和出度之间的最大差值\(T_ {N} \) 。甲subdigraph ħ圆弧各色有向图的d,如果被称为彩虹每对圆弧在ħ有不同的颜色。在本文中,我们表明,每个顶点v而圆弧有色赛\(T_ {N} \)\(\ varDelta ^ { -周一}(T_n)\乐\ varDelta ^ {+周一}(T_n)\ )至少包含在\(\ frac {\ delta(v)(n- \ delta(v)-i(T_n))} {2}-[\ varDelta ^ {-mon}(T_ {n})( n-1)+ \ varDelta ^ {+ mon}(T_ {n})d ^ +(v)] \)彩虹三角形,其中\(\ delta(v)= \ min \ {d ^ +(v), d ^-(v)\} \)。我们还为\(T_ {n} \)给出了一些最大单色度条件,以包含彩虹三角形,并包含通过给定顶点的彩虹三角形。最后,我们提供一些示例,表明我们结果中的某些条件是最可能的。

更新日期:2021-05-14
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