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 χom(G). Here properties of open monophonic chromatic number of connected graphs are studied. Open monophonic chromatic number of some standard graphs are identified. For ۳≤ m ≤n, there is a connected graph G such that χ(G)= m and χom(G)=n. For ۳≤ m ≤n, there is a connected graph Gsuch that om(G)=m and χ(G)= χom(G)=n. Let r, d be two integers such that r< d ≤ ۲r and suppose k≥ ۲. Then there exists a connected graph G with rad(G) = r, diam(G) = d and χom(G)=k.

Keywords:

Chromatic set , Chromatic number , Open Monophonic number , Open Monophonic chromatic number