Quasi Neighborhood Balanced Coloring of Graphs

TL;DR

This paper introduces the concept of quasi neighborhood balanced coloring in graphs, explores its variants, and proves the NP-completeness of recognizing such colorings.

Contribution

It defines new coloring variants, provides examples of graph classes that admit them, and establishes the computational complexity of the recognition problem.

Findings

Several graph classes admit quasi neighborhood balanced colorings.

Recognition of such colorings is NP-complete.

No forbidden subgraph characterization exists for this class.

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.