Abstract
A proper coloring on [Formula: see text] is called a [Formula: see text]-coloring if every color class has a vertex [Formula: see text] such that [Formula: see text] has at least one neighbor from all other color classes. The maximum integer [Formula: see text] for which [Formula: see text] admits a [Formula: see text]-coloring with [Formula: see text] colors is called the [Formula: see text]-chromatic number of [Formula: see text]. Another interesting coloring parameter is the neighborhood pseudo chromatic number, denoted by [Formula: see text]. It is the maximum number of colors used for a pseudo coloring of [Formula: see text] such that each vertex [Formula: see text] has at least two vertices in [Formula: see text] which receive the same color. Motivated by the extensive research and applications in the area of [Formula: see text]-coloring and pseudo coloring, we introduce a new parameter called the neighborhood [Formula: see text]-pseudo chromatic number and obtain an interesting characterization to find the neighborhood [Formula: see text]-pseudo chromatic number of an arbitrary graph.
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