On open distance pattern coloring of graphs

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Given any arbitrary nonempty subset [Formula: see text] of vertices in a graph [Formula: see text], each vertex [Formula: see text] in [Formula: see text] is associated with the vertex [Formula: see text] and is called an [Formula: see text]-open distance pattern coloring (odp coloring) of [Formula: see text] if no two adjacent vertices receive the same color [Formula: see text]. If such an [Formula: see text] exists, for a graph [Formula: see text], then [Formula: see text] is called an Open Distance Pattern Colorable (odpc) graph. The minimum cardinality of an odp coloring set in [Formula: see text], if it exists, is called the odpc-number of [Formula: see text] and is denoted by [Formula: see text]. Given any property [Formula: see text], we establish the characterization of odpc-graph with the property [Formula: see text]. Some classes of graphs are proved to be odpc. It is also proved that for a given positive integer [Formula: see text], there exists an odpc graph with an odp coloring set of cardinality [Formula: see text].