发表机构
Università Cattolica del Sacro Cuore（天主教撒克罗库雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究给定共度序列是否可由某个简单图实现的问题，推广了Erdős-Gallai定理和友谊定理，并针对平面C4-自由图类给出了完整刻画与构造方法。
AI 中文摘要
给定一个在 $v$ 个顶点上的简单图 $G = (V, E)$ 以及两个不同的顶点 $x, y \in V$，与顶点对 $\{x, y\}$ 关联的共度 $c_{x,y}$ 是它们在图 $G$ 中公共邻居的数量。图 $G$ 的共度序列，记为 $\gamma(G)$，是与所有可能的不同顶点对关联的所有共度的列表，按非递增顺序排列。在本文中，我们考虑以下问题，该问题可以看作是 Erdős 和 Gallai 的一个结果以及 Erdős、Rényi 和 Sós 的友谊定理的推广：给定一个整数 $v\geqslant 2$ 和一个按非递增顺序排列的非负整数序列 $\gamma$，确定是否存在一个在 $v$ 个顶点上的简单图，其共度序列为 $\gamma$，并且在肯定回答的情况下，提供这样的图。我们针对平面 $C_4$-自由图类给出了这个问题的完整答案。
英文摘要
Given a simple graph $G = (V, E)$ on $v$ vertices and two distinct vertices $x, y \in V$, the co-degree $c_{x,y}$ associated to the pair $\{x, y\}$ is the number of their common neighbors in the graph $G$. The co-degree sequence of $G$, denoted by $γ(G)$, is the list of all the co-degrees associated to all the possible pairs of distinct vertices, arranged in non-increasing order. In this paper we consider the following problem, which can be viewed as a generalization of a result by Erdős and Gallai as well as of the Erdős, Rényi and Sós' friendship theorem: given an integer $v\geqslant 2$ and a sequence $γ$ of nonnegative integers arranged in non-increasing order, establish if there exists a simple graph on $v$ vertices having $γ$ as its co-degree sequence and, in case of positive answer, provide such a graph. We provide a full answer to this problem for the class of planar $C_4$-free graphs.
