Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs

For finite sequence $\underset{‾}{\text{em}\text{d}}$ of positive integers, we consider graphs that have $\underset{‾}{\text{em}\text{d}}$ as their list of vertex degrees, and bipartite graphs for which each part has $\underset{‾}{\text{em}\text{d}}$ as its list of vertex degrees. In particular, we make a connection between a result for bipartite graphs by Alon, Ben-Shimon and Krivelevich and a result of Zverovich and Zverovich for graphs, and we give an improvement of a result of Zverovich and Zverovich. We show that the bipartite graphs with vertex degree sequences $(\underset{‾}{\text{em}\text{d}},\underset{‾}{\text{em}\text{d}})$ are in one to one correspondence with graphs with loops with reduced degree sequence $\underset{‾}{\text{em}\text{d}}$, where the reduced degree of a vertex is defined to be the number of edges incident to the vertex, with loops counted only once. We also give two Erd\H{o}s–Gallai type theorems for graphs with loops.