The Split Chromatic Sets in Graphs

In this paper, we initiate to study the concepts of split chromatic sets in a connected graph $G$ and an associated variable, called the split chromatic number of $G$. The concept of chromatic set C and $k$-chromatic graph $G_k$ was introduced in [4, 5]. A set C $\subseteq V(G)$ of vertices in a connected graph $G$ is said to be a split chromatic set of $G$ if C is a chromatic set of $G$ and $\langle V \sim \mathrm{C} \rangle$ is disconnected. The minimum cardinality among all the split chromatic sets of $G$ is called the split chromatic number of $G$ and is notated by $\chi_s(G)$. The split chromatic number of some standard graphs are identified. In this work, introduce two non-disjoint families of graphs, namely, chromatic (split chromatic) graph family. Next, realizes the chromatic (split chromatic) graph family by providing its chromatic (split chromatic) number and chromatic (split chromatic) family index of the graphs respectively.

1. For each $(m, n) \in Z^+ \times Z^+$ with ordering $2 \le m \le n$, there exists $G \in CS[G]_m$ such that $\chi_s(G) = m$ and SCI pair = $(m, n)$

2. For $p, q, r \in Z^+$ with ordering $2 \le p \le q \le 2p$ and $r > q + 2$, there exists $G \in C[G]_{p+1}$ such that $\chi_s(G) = q + 1$ and $|V(G)| = r$.

3. For each $(m, n, r) \in Z^+ \times Z^+ \times Z^+$, there exists $G \in CS[G]_m$ such that $\chi_s(G) = n$, SCI pair = $(m, n)$ and $|V(G)| = r$, where $3 \le m \le n < r$ and $r > n + 2$