New Invariants for Partitioning a Graph Into 2‐Connected Subgraphs

ABSTRACT Let be a graph of order . For an integer , a partition of is called a ‐proper partition of if every induces a ‐connected subgraph of . This concept was introduced by Ferrara et al., and Borozan et al. gave minimum degree conditions for the existence of a ‐proper partition. In particular, when , they proved that if , then has a 2‐proper partition with . Later, Chen et al. extended the result by giving a minimum degree sum condition for the existence of a 2‐proper partition. In this paper, we introduce two new invariants of graphs and , which are defined from degree sum of particular independent sets. Our result is that if , then with some exceptions, has a 2‐proper partition with . We completely determine exceptional graphs. This result implies both of results by Borozan et al. and by Chen et al. Moreover, we obtain a minimum degree product condition for the existence of a 2‐proper partition as a corollary of our result.