Starting with the Criterion by Gutman and Ruch for graphical partitions, Gutman analyzed degree sequences of connected graphs with cyclomatic number $c$, for $c\le 5$. In this paper, his results are revisited and, based on the Erdös-Gallai Criterion, extended to arbitrary values of $c$. Necessary and sufficient conditions are obtained for any partition to be the degree sequence of a connected graph with cyclomatic number $c$.