平面三角剖分图中非连通图的Anti-Ramsey数

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平面三角剖分图中非连通图的Anti-Ramsey数
The Anti-Ramsey Number of Unconnected Graphs in Plane Triangulation Graphs

DOI: 10.12677/AAM.2023.126304, PDF, HTML,
作者: 罗冬连^*, 顾俊琪：浙江师范大学数学科学学院，浙江金华
关键词: 彩虹匹配；Anti-Ramsey数；平面三角剖分；Rainbow Matching； Anti-Ramsey Number； Plane Triangulations

摘要: 给定图 G 的一个边染色，如果图 G 的任意两条边颜色都不相同, 那么就说图 G 是彩虹的。图 H 在图 G 中的 anti-Ramsey 数是使得边染色图 G 中不存在任何彩虹子图 H 的最大颜色数。图的 anti-Ramsey 数目前得到广泛的研究, 尤其是匹配在多种图类中的 anti-Ramsey 数得到广泛而深入的研究。 Gilboa 和Roditty 研究了由小的连通分支构成的图在完全图中的 anti-Ramsey 数，而非连通图在平面图中的 anti-Ramsey 数除匹配外结果较少。本论文将继续以这个方向研究边染色图中 C ₃ ∪ tP ₂ 这个非连通图在平面三角剖分图中的 anti-Ramsey 数，得到了对任意n ≥ 2t + 3, t ≥ 2, 2n + 3t − 9 ≤ AR(T _n, C ₃ ∪ tP ₂) ≤ 2n + 4t − 5。

Abstract: Given an edge-coloring of a graph G, G is said to be rainbow if any two edges of G receive different colors. Given two graphs G and H, the anti-Ramsey number of H in G is defined to be the maximum number of colors in an edge-colored graph G which contains no rainbow copies of H. The anti-Ramsey numbers for graphs, especially matchings, have been studied in several graph classes. Gilboa and Roditty focused on the anti-Ramsey number of graphs with small components, but the results of the anti-Ramsey number of graphs with small components in plane graph are few. In this paper, we continue the work in this direct and determine the anti-Ramsey number of C ₃ ∪ tP ₂ in plane triangulations, then we can get 2n + 3t − 9 ≤ AR( T _n , C ₃ ∪ tP ₂) ≤ 2n + 4t − 5 for n ≥ 2t + 3, t ≥ 2.

文章引用：罗冬连, 顾俊琪. 平面三角剖分图中非连通图的Anti-Ramsey数[J]. 应用数学进展, 2023, 12(6): 3030-3038. https://doi.org/10.12677/AAM.2023.126304

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