Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs

TL;DR

This paper explores graphical degree sequences for graphs and bipartite graphs, establishing new connections, improving existing theorems, and providing Erdős–Gallai type results for graphs with loops.

Contribution

It links bipartite graph degree sequences with graphs with loops and offers improved theorems for graphical sequences, advancing understanding of graphical degree conditions.

Findings

Established a one-to-one correspondence between bipartite graphs with degree sequence (d,d) and graphs with loops with degree sequence d.

Provided two Erdős–Gallai type theorems for graphs with loops.

Connected results for bipartite graphs to those for graphs with loops, improving previous bounds.

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 Erd\H{o}s--Gallai type theorems for graphs with loops.