In chess, a knight's move consists of two spaces either vertically or horizontally, followed by one space in the perpendicular direction. In this way, every knight's move results in an L shaped displacement from the original position. To model all of the legal knight's moves on an n x n chessboard, we can create a graph where each vertex represents a square on the board, and two vertices are adjacent if an only if a knight can legally move between the corresponding squares. Use the handshake lemma to determine the number of edges in GK_n
Is GK_n always, sometimes or never Eulerian
Does GK_n always, sometimes or never contain an Euler trail
By use of the Handshake Lemma edges are twice the amount of degree sum so if you had a graph GK_4 with 16 vertices, it would have degree sum 48 and 24 edges. I'm not sure how to use this information to determine the number of edges in GK_n. The abstractness is what confuses me.