论小属曲面中 1- 嵌入图的受限匹配扩展

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-07-22 DOI:10.1016/j.disc.2024.114172

Jiangyue Zhang, Yan Wu, Heping Zhang

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引用次数: 0

摘要

设是一个至少有顶点的连通图，且包含一个完美匹配。那么，对于每一对大小分别为和的不相邻匹配，都存在一个完美匹配，使得和。如果可以在 Σ 中画出一条边，使得每条边最多与另一条边相交于一点，则该图位于曲面 Σ 中。R.E.L. Aldred 和 M.D. Plummer 研究了嵌入平面、环面、投影面和克莱因瓶中的图的性质。在本文中，我们研究了小属的曲面中 1-embeddable 图形的性质。结果表明，平面或投影面中没有 1-embeddable 图形，环面或克莱因瓶中也没有 1-embeddable 图形。作为推论，平面或投影面中没有 1- 嵌入图是 5- 可扩展的，环面或克莱因瓶中没有 1- 嵌入图是 6- 可扩展的。一些例子表明，这样的结果是最有可能的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On restricted matching extension of 1-embeddable graphs in surfaces with small genus

Let G be a connected graph with at least $2 (m + n + 1)$ vertices that contains a perfect matching. Then G is $E (m, n)$ if for each pair of disjoint matchings $M, N \subseteq E (G)$ of size m and n, respectively, there exists a perfect matching F in G such that $M \subseteq F$ and $F \cap N = \emptyset$ . A graph G is 1-embeddable in a surface Σ if G can be drawn in Σ so that every edge of G crosses at most one other edge at a point. R.E.L. Aldred and M.D. Plummer [1], [2] investigated the properties $E (m, n)$ for graphs embedded in the plane, torus, projective plane and Klein bottle. In this paper, we study the property $E (m, n)$ for 1-embeddable graphs in surfaces with small genus. It is shown that no 1-embeddable graph in the plane or projective plane is $E (4, 1)$ and no 1-embeddable graph in the torus or Klein bottle is $E (5, 1)$ . As corollaries, no 1-embeddable graph in the plane or projective plane is 5-extendable and no 1-embeddable graph in the torus or Klein bottle is 6-extendable. Some examples show that such results are best possible.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.