On the Universality and Extremality of graphs with a distance constrained colouring

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance two must receive distinct integers. The lambda chromatic number (or the $\lambda$ number) of a graph $G$ is the least positive integer among all the maximum assigned positive integer over all possible lambda colouring of the graph $G$. Here we have primarily shown that every graph with lambda chromatic number $t$ can be embedded in a graph, with lambda chromatic number $t$, which admits a partition of the vertex set into colour classes of equal size. It is further proved that if an $n-$vertex graph with lambda chromatic number $t\geq5$, where $n\geq t+1$, contains maximum number of edges, then the vertex set of such graph admits an equitable partition. For such an admitted equitable partition there are either $0$ or $\min\{|A|,|B|\}$ number of edges between each pair $(A,B)$ of subsets (i.e. roughly, such partition is a "sparse like" equitable partition). Here we establish a classification result, identifying all possible $n-$vertex graphs with lambda chromatic number $t\geq3$, where $n\geq t+1$, which contain maximum number of edges. Such classification provides a solution of a problem posed more than two decades ago by John P. Georges and David W. Mauro.