图的友好性——极值与保持者
Graph Cordiality -- Extremes and Preservers
AI总结：
本文研究了无向图的cordial性与(2,3)-可定向性，给出了最小反例及边数上界，并证明了强保持这些性质的线性算子必为顶点置换。
摘要：
如果一个无向图的顶点存在友好的(0,1)-标号，且该标号能诱导出边的友好(0,1)-标号，则称该无向图是cordial的。如果无向图$G$的顶点存在友好的(0,1)-标号，使得大约三分之一的边与标号相同的顶点关联，则称$G$是(2,3)-可定向的。也就是说，存在某个有向图，它是$G$的一个定向，且是(2,3)-cordial的。本文给出了最小的非cordial/非(2,3)-可定向图的例子，并提出了cordial/(2,3)-可定向图中可能边数的上界。此外，还证明了如果$T$是作用于所有$n$个顶点的无向图集合上的线性算子，且强保持cordial图集合或(2,3)-可定向图集合，则$T$是一个顶点置换。
英文摘要：
An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph $G$ is said to be $(2,3)$-orientable if there exists a friendly (0,1)-labeling of the vertices of $G$ such that about one third of the edges are incident to vertices labeled the same. That is, there is some digraph that is an orientation of $G$ that is $(2,3)$-cordial. Examples of the smallest noncordial/non-$(2,3)$-orientable graphs are given and upper bounds on the possible number of edges in a cordial/$(2,3)$-orientable graph are presented. It is also shown that if $T$ is a linear operator on the set of all undirected graphs on $n$ vertices that strongly preserves the set of cordial graphs or the set of $(2,3)$-orientable graphs then $T$ is a vertex permutation..
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
