the open monophonic chromatic number of a graph

A set p of vertices in a connected graph g is called open monophonic chromatic set if p is both an open monophonic set and a chromatic set. the minimum cardinality among the set of all open monophonic chromatic sets is called open monophonic chromatic number and is denoted by chi;om(g). here properties of open monophonic chromatic number of connected graphs are studied. open monophonic chromatic number of some standard graphs are identified. for 3 le; m le;n, there is a connected graph g such that chi;(g)= m and chi;om(g)=n. for 3 le; m le;n, there is a connected graph gsuch that om(g)=m and chi;(g)= chi;om(g)=n. let r, d be two integers such that r lt; d le; 2r and suppose k ge; 2. then there exists a connected graph g with rad(g) = r, diam(g) = d and chi;om(g)=k.