Proper 2-coloring game on some trees

Abstract

Let Pn denote the path with exactly n vertices and let Tn,i denote the tree obtained from Pn−1 by adding a pendant vertex to the (i+1)-th vertex of Pn−1, where n≥4 and 1≤i≤n−3. In this paper we study proper 2-coloring game on Tn,i by using Sprague–Grundy function and give the optimal strategy of proper 2-coloring game on Tn,i.