Perfect Divisions in (P₃ ∪ P₄, P₆,Bull)-Free Graphs

A graph G is said to be perfect if (Formula presented.) for every induced subgraph H of G, where (Formula presented.) and (Formula presented.) denote the clique number and the chromatic number of H. We say that a graph G admits a perfect division if its vertex set can be partitioned into two subsets A and B such that (Formula presented.) is perfect and (Formula presented.). If every induced subgraph of G admits a perfect division, then G is called perfectly divisible. A graph (Formula presented.) is the disjoint union of paths (Formula presented.) and (Formula presented.). A bull refers to the graph consisting of a triangle with two disjoint pendant edges. A homogeneous set X is a proper subset of (Formula presented.) with at least two vertices such that every vertex in (Formula presented.) is either complete or anticomplete to X. In this paper, we prove that every ( (Formula presented.), bull)-free graph G with (Formula presented.) admits a perfect division, provided that G contains no homogeneous set. Moreover, we establish that this clique number condition is tight by presenting a counterexample with clique number of exactly 2.