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arXiv · 1302.3657

Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs

Grant Cairns·Stacey Mendan
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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.

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Grant Cairns, Stacey Mendan. 2013-02-15. Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs. https://arxiv.org/abs/1302.3657

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