  A celebrated theorem of Stiebitz asserts that any graph with minimum degree
at least $s+t+1$ can be partitioned into two parts which induce two subgraphs
with minimum degree at least $s$ and $t$, respectively. This resolved a
conjecture of Thomassen. In this paper, we prove that for $s,t\geq 2$, if a
graph $G$ contains no cycle of length four and has minimum degree at least
$s+t-1$, then $G$ can be partitioned into two parts which induce two subgraphs
with minimum degree at least $s$ and $t$, respectively. This improves the
result of Diwan, who proved the same statement for graphs of girth at least
five. Our proof also works for the case of variable functions, in which the
bounds are sharp as showing by some polarity graphs. As a corollary, it follows
that any graph containing no cycle of length four with minimum degree at least
$k+1$ contains $k$ vertex-disjoint cycles.

Combinatorics