Quasi Neighborhood Balanced Coloring of Graphs
Maurice Geneveiva Almeida
TL;DR
This work introduces quasi neighborhood balanced coloring (QNBC) and investigates its structural and algorithmic properties. It derives parity-based counting relations and congruences, and catalogs several graph families that admit QNBCs, including variants such as uniform, positive, and negative. It demonstrates the preservation of QNBC under multiple graph products and joins, and proves the class is non-hereditary, ruling out a forbidden-subgraph characterization. The decision problem for QNBC is shown to be NP-complete via a reduction from Neighborhood Balanced Coloring, highlighting intrinsic computational difficulty. Together, these results deepen understanding of equitable neighbor-color distributions and their behavior under standard graph constructions.
Abstract
For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors differ by at most one. In addition, there must be at least one vertex in G for which this difference is exactly one. If a graph G admits such a colouring, then G is said to be a quasi-neighbourhood balanced colored graph. We also define variants of such a coloring, like uniform quasi neighborhood balanced coloring, positive quasi neighborhood balanced coloring and negative quasi neighborhood balanced coloring based on the color of the extra neighbor of every vertex of odd degree of the graph G. We present several examples of graph classes that admit the various variants of quasi neighborhood balanced coloring. We also discuss various graph operations involving such graphs. Furthermore, we prove that there is no forbidden subgraph characterization for the class of quasi neighborhood balanced coloring and show that the problem of determining whether a given graph has such a coloring is NP-complete.