Weights of induced subgraphs in K1, r-free graphs

Let H be a subgraph of a given graph G . The weight w ( H ) is defined to be the degree sum of the vertices of H in G . Investigations of this parameter are initiated by the result of Kotzig in 1955 who proved that every 3-connected planar graph contains an edge of weight at most 13.In this paper, we seek a bound f depending on some parameters of G and H such that w ( H ' ) ? f for every induced subgraph H ' in G isomorphic to H . We obtain the following result for r ? 3 : If H is an induced k -colorable subgraph of a K 1 , r -free graph G , and I ? is a largest independent set in G , then w ( H ) ? k ( r - 1 ) ( n - α ( G ) ) - ? v ? V ( H ) - I ? ( ( k - 1 ) ( r - 1 ) - d H ( v ) ) .Moreover, we give some sharpness examples.