Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs

Abstract

For finite sequence $\underbar{\em d}$ of positive integers, we consider graphs that have $\underbar{\em d}$ as their list of vertex degrees, and bipartite graphs for which each part has $\underbar{\em d}$ 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 $(\underbar{\em d},\underbar{\em d}\,)$ are in one to one correspondence with graphs with loops with reduced degree sequence $\underbar{\em d}$, 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 Erdős--Gallai type theorems for graphs with loops.

Paper has been rewritten for publication as two separate papers: one on graphs with loops and one on the improvement of the Zverovich--Zverovich result