将投影面划分为两个入射丰富的部分

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2024-10-06 DOI:10.1002/jcd.21956

Zoltán Lóránt Nagy

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引用次数: 0

摘要

图 G $G$ 的顶点集 V ( G ) $V(G)$的内部或友好分区是对两个非空集 A ∪ B $A\cup B$ 的分区，使得每个顶点在自己的类中至少有和在另一个类中一样多的邻居。受 Diwan 关于高周长图内部分区存在性证明的启发，我们给出了投影平面入射图内部分区存在性的构造性证明，并讨论了其几何性质。此外，我们还精确地确定了对于平方阶的德萨格平面的入射图，所有顶点同时可以达到的本类邻集与他类邻集的最大可能差值。

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Partitioning the projective plane into two incidence-rich parts

An internal or friendly partition of a vertex set $V (G)$ of a graph $G$ is a partition to two nonempty sets $A \cup B$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.