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 K1,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)−dH(v)).Moreover, we give some sharpness examples.