Closed Neighborhood Balanced Coloring of Graphs

A coloring of the vertex set of a graph using the colors red and blue is a closed neighborhood balanced coloring if for each vertex there are an equal number of red and blue vertices in its closed neighborhood. A graph with such a coloring is called a CNBC graph. Freyberg and Marr studied the related class of NBC graphs where closed neighborhood is replaced by open neighborhood. We prove results about CNBC graphs and NBC graphs. We show that the class of CNBC graphs is not hereditary, that the sizes of the color classes can be arbitrarily different, and that if the sizes of the color classes are equal, then a graph is a CNBC graph if and only if its complement is an NBC graph. When the sizes of the color classes are equal, we show that the join of two CNBC graphs is a CNBC graph, and the lexicographic product of a CNBC graph with any graph is a CNBC graph. We prove that the Cartesian product of any CNBC graph and any NBC graph is a CNBC graph, and characterize when a hypercube is an NBC graph or a CNBC graph, but show that the product of two CNBC graphs need not be an NBC graph. We show that the strong product of a CNBC graph with any graph is a CNBC graph. We construct infinite families of circulants that are CNBC graphs, and give characterizations of CNBC trees, generalized Petersen graphs, cubic circulants and quintic circulants when \(n\equiv 2 \pmod 4\).