Prove the following related to graph theory: Show that an edge in a simple graph is a cut edge if and only if this edge is not part of any simple circuit in the graph. Show that in any simple graph there is a path from any vertex of odd degree to some other vertex of odd degree. Given below the initial chessboard configuration, prove of disprove whether the other two boards can be independently generated by moving one knight (W)hite, (B)lack) at a time in finite number of steps? Then show that there is no knight's tour on one of the chessboards. (exhaustive search is forbidden), (knight's tour is a sequence of L-moves by a knight starting at some square and visiting each square once)