Sparse regular induced subgraphs in 2P32P3-free graphs

We call graphs of a fixed degree kksparse regular graphs and their complements dense regular graphs  . Recently, it was conjectured that finding a maximum regular induced subgraph HH in a 2P32P3-free graph can be solved in polynomial time if and only if HH is sparse. In the present paper, we prove the “sparse” part of this conjecture, i.e., we show that for each fixed kk, the problem of finding a maximum kk-regular induced subgraph in a 2P32P3-free graph can be solved in polynomial time.