... and w with the same color. Now the vertex v between u, w on P4is not inC, but since u, w areatdistanceatleast3onC the path uvw together with one of theu, w-segments of C yields a shorter non-contractible ... preliminariesSuppose F is a nonempty family of connected bipartite graphs, each with at least 3vertices. An r-coloring of a graph G is said to be F-free if G contains no 2-colored subgraphisomorphic ... otherwise it would contain a 2-colored P4). We will use the following terminology.Definition 2.1. An r-coloring of G is called a star coloring if there are no 2-colored pathson 4 vertices....