Peg solitaire on graphs with large maximum degree

Abstract

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.