On the Systems of Distinct Representatives for the Family of Sets Formed by Neighborhoods of Vertices

Currently, the theory of systems of distinct representatives is being carefully examined and reworked, often in a more general context. With time, considerable literature has grown, and new theories are proposed that require the system of distinct representatives to possess additional properties. In this article, one such context is considered. A graph is completely defined by the relationship between its vertices, also known as the neighborhood of vertices in graph theoretical context. Several parameters are defined based on the set of all neighborhoods of vertices and are explored. This article delves into the well-known problem of the existence of a system of distinct representatives applied for the family of neighborhoods. That is, for a graph G and a family F of subsets of V(G) formed by the neighborhoods of vertices, the necessary and sufficient condition to have at least one system of distinct representatives is derived. A new type of derived graph called the neighborhood distinct representative graph is introduced, through which the article relates the existence of unique representatives with properties like connectedness, acyclicity, etc. Further, an expression for the number of systems of distinct representatives for the above family is derived. Consequently, the expression for the same is derived for some standard classes of graphs, like trees, cycles, wheel graphs, fan graphs, etc.