I consider an infinite Y-shaped graph which is obtained by

connecting three half lines at a point

and study the Allen-Cahn equation in the graph. I consider the existence of a solution which behaves like a travelling solution in a line which exists for all the time with a stability analysis. This presentation is based on joint work with Prof. Y. Morita (Ryukoku univ.) and Mr. Takazawa (Hokkaido univ.).