On connected partition with degree constraints

Let s and t be nonnegative integers, and let f(s,t) be the smallest positive integer such that every f(s,t)-connected graph G admits a vertex partition (S,T) such that G[S] is s-connected and G[T] is t-connected. An (s,t)-feasible partition of graph G is a partition (S,T) of V(G) such that the minimum degree of G[S] and G[T] is at least s and t, respectively, and an (s,t)-feasible partition is said to be connected if both G[S] and G[T] are connected. Let g(s,t) be the smallest positive integer such that every graph G with minimum degree at least g(s,t) admits an (s,t)-feasible partition. Thomassen conjectured that f(s,t)=s+t+1 and g(s,t)=s+t+1. The later conjecture was confirmed by Stiebitz, while the former one is still open. In this paper, we improve Hajnal's upper bound f(s,t)≤4s+4t−13 by showing that f(s,t)≤⌈19(s−1)6⌉+⌈19(t−1)6⌉+1 if min⁡{s,t}≥3. We also show that any connected graph with an (s,t)-feasible partition also admits a connected (s,t)-feasible partition. This implies that we can extend all the results of [4,5,7–9,11,16] to connected (s,t)-feasible partitions. Finally, we show that if min⁡{s,t}≥2, then g(s,t)≤s+t−1 for {K3,K2,3,H1,H2,H3}-free graphs, and g(s,t)≤s+t for K2,3+-free graphs, where K2,3+ denote the set of graphs obtained from K2,3 by adding exactly one edge joining its two vertices, and H1, H2 and H3 are three specific graphs defined in the paper.