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.
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Funding
W. Li is supported by NSFC (nos. 11301098 and 11601428) and GXNFS (no. 2016GXNSFFA380011); S. Zhang is supported by NSFC (nos. \(\sim \)12071370 and U1803263); Y. Bai is supported by NSFC (no. 11601430), Natural Science Foundation of Shaanxi Province (No. 2020JQ-099), the Fundamental Research Funds for the Central Universities (no. 3102019ghjd003) and China Postdoctoral Science Foundation (no. 2016M590969) and R. Li is supported by Natural Science Foundation of Shaanxi Province (No. 2020JQ-111).
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Li, W., Zhang, S., Bai, Y. et al. Rainbow Triangles in Arc-Colored Tournaments. Graphs and Combinatorics 37, 1271–1290 (2021). https://doi.org/10.1007/s00373-021-02313-w
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DOI: https://doi.org/10.1007/s00373-021-02313-w