  In this paper, we introduce the concept of curling subsequence of simple,
finite and connected graphs. A curling subsequence is a maximal subsequence $C$
of the degree sequence of a simple connected graph $G$ for which the curling
number $cn(G)$ corresponds to the curling number of the degree sequence per se
and hence we call it the curling number of the graph $G$. A maximal degree
subsequence with equal entries is called an identity subsequence. The number of
identity curling subsequences in a simple connected graph $G$ is denoted
$ic(G).$ We show that the curling number conjecture holds for the degree
sequence of a simple connected graph $G$ on $n \geq 1$ vertices. We also
introduce the notion of the compound curling number of a simple connected graph
$G$ and then initiate a study on the curling number of certain standard graphs
like Jaco graphs and set-graphs.
