Weights of induced subgraphs in -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. s 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 ) ) . er, we give some sharpness examples.