The independence number and chromatic number in a reduced graph

Let G be a loopless connected graph. A graph G is reduced if it contains no collapsible subgraph. Catlin (posted by Chen and Lai [9]) conjectured that every connected reduced graph is either 2-colorable or 3-colorable. A weaker conjecture states that the independence number of a connected reduced graph G is at least one-third of its number of vertices. In this paper, we establish a lower bound on the independence number in reduced graphs. As an application, we examine the independence number conjecture for reduced graphs with a given upper bound on the number of vertices. Also, we investigate the chromatic number of reduced planar graphs under given conditions.