Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of \(K_n\) contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the 2-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.

中文翻译：

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两类单环图的Ramsey和Gallai-Ramsey数

给定一个图G和一个正整数k，将Gallai-Ramsey数定义为最小顶点数n，使得\（K_n \）的任何k边缘着色都包含彩虹（所有不同颜色的）三角形或单色G的副本。在本文中，我们考虑两类单圈图，具有额外边缘的星形和位于一端的三角形路径。我们为这两类图提供2色Ramsey数，并使用它们获得Gallai-Ramsey数的一般上限和下限。