Generalized chromatic numbers and additive hereditary properties of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let and be additive hereditary properties of graphs. The generalized chromatic number ${\displaystyle {\chi }_{}\left(\right)}$ is defined as follows: ${\displaystyle {\chi }_{}\left(\right)=n}$ iff ⊆ ⁿ but ${\displaystyle {⊊}^{n-1}}$. We investigate the generalized chromatic numbers of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ and ₖ.