To build a bipartite matching model for this problem, we can create a network where each node represents a square on the chessboard and arcs represent the possible movements of the knights. We will divide the nodes into two sets: one set for the black squares and another set for the white squares. This division ensures that the network is bipartite since a knight on a black square can only move to a white square and vice versa.

Here's how to construct the network:

Create a node for each square on the chessboard.

Divide the nodes into two sets: black and white.

Draw an arc from a black node to a white node if a knight on the black square can move to the white square.

Draw an arc from a white node to a black node if a knight on the white square can move to the black square.

Now, we need to determine the maximum size of a valid arrangement for an m x n board. A valid arrangement is a matching in the bipartite graph, where each node is matched to at most one other node, and each knight covers one additional square. The maximum size of a valid arrangement is the maximum number of arcs in a matching.

To find the maximum matching, we can use a maximum flow algorithm, such as the Ford$-$Fulkerson algorithm, on the residual network. The residual network is constructed by adding reverse arcs for each arc in the original network and setting the capacity of each arc to $1.$ The maximum flow in the residual network is equal to the size of the maximum matching.

Final answer: To determine the maximum size of a valid arrangement for an m x n board, construct a bipartite graph with nodes representing squares and arcs representing possible knight movements. Divide the nodes into black and white sets. Use a maximum flow algorithm on the residual network to find the maximum matching, which represents the maximum size of a valid arrangement.