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.