On partitions of graphs under degree constraints

Let s,t be two integers, and let g(s,t) denote the minimum integer such that the vertex set of a graph of minimum degree at least g(s,t) can be partitioned into two nonempty sets which induce subgraphs of minimum degree at least s and t, respectively. In this paper, it is shown that, (1) for positive integers s and t, g(s,t)≤s+t on (K4−e)-free graphs except K3, and (2) for integers s≥2 and t≥2, g(s,t)≤s+t−1 on triangle-free graphs in which no two quadrilaterals share edges. Our first conclusion generalizes a result of Kaneko (1998), and the second generalizes a result of Diwan (2000).