A Characterization of the Degree Sequence of the Graph with Cyclomatic Number \(k\)

Abstract

Let \((d_1, d_2, \dots, d_n)\) be a sequence of positive integers with \(n-1 \geq d_1 \geq d_2 \geq \dots \geq d_n\). We give a characterization of \((d_1, d_2, \dots, d_n)\) that is the degree sequence of a graph with cyclomatic number \(k\). This simplifies the characterization of Erdős-Gallai.