Exact 2-Distance b-Coloring and Exact 2-Distance b-Continuity of Helm Graph ????????

An exact 2-distance coloring of a graph ???? is a coloring of vertices of ???? such that any two vertices which are at distance exactly 2 receive distinct colors. An exact 2-distance chromatic number????2(????) of ???? is the minimum ???? for which ???? admits an exact 2-distance coloring with ???? colors. A ????-coloring of ???? by ???? colors is a proper ????-vertex coloring such that in each color class, there exists a vertex called a color dominating vertex which has a neighbor in every other color class. A vertex that has a 2-neighbor in all other color classes is called an exact 2-distance color dominating vertex (or an ????2-cdv). Exact 2-distance ????-coloring (or an ????2????-coloring) of ???? is an exact 2-distance coloring such that each color class contains an ????2- cdv. An exact 2-distance ????-chromatic number (or an ????2????-number) ????2????(????) of ???? is the largest integer ???? such that ???? has an ????2????-coloring with ????colors. If for each integer????, ????2(????) ≤ ???? ≤ ????2????(????), ???? has an ????2????-coloring by ???? colors, then ???? is said to be an exact 2-distance ????- continuous graph. In this paper, the ????2????-number????2????(????????)of the helm graph ????????is obtained and ????2????-continuity of ????????is discussed.