最小k因子临界平面图

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2026-10-12 Epub Date: 2026-06-13 DOI:10.1002/jgt.70083

Qiuli Li, Fuliang Lu, Heping Zhang

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A <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n </semantics></math>-factor-critical graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> is minimal if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math> is not <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n </semantics></math>-factor-critical for any edge <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math> in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math>. Favaron and Shi posed the conjecture that every minimal <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n </semantics></math>-factor-critical graph is of minimum degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n </semantics></math> in 1998. In this paper, we confirm the conjecture for planar graphs.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 4","pages":"545-552"},"PeriodicalIF":1.0000,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Minimal \\n \\n \\n \\n k\\n \\n \\n -Factor-Critical Planar Graphs\",\"authors\":\"Qiuli Li, Fuliang Lu, Heping Zhang\",\"doi\":\"10.1002/jgt.70083\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n A graph of order <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> is said to be <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>-factor-critical (<math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mn>0</mn>\\n \\n <mo>≤</mo>\\n \\n <mi>k</mi>\\n \\n <mo><</mo>\\n \\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>) if the removal of any <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> vertices results in a graph with a perfect matching. 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Favaron and Shi posed the conjecture that every minimal <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>-factor-critical graph is of minimum degree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n </semantics></math> in 1998. 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引用次数: 0

摘要

一个n阶的图被称为k因子临界图（0≤k）< n)如果移除任意k个顶点会得到一个完美匹配的图。如果G−e，则k因子临界图G最小对于G中的任何边e都不是k因子临界的。Favaron和Shi在1998年提出了每一个最小k -关键因子图都是最小k + 1度的猜想。本文证实了平面图的这个猜想。

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 Minimal k -Factor-Critical Planar Graphs

A graph of order $n$ is said to be $k$ -factor-critical ( $0 \leq k < n$ ) if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$ -factor-critical graph $G$ is minimal if $G - e$ is not $k$ -factor-critical for any edge $e$ in $G$ . Favaron and Shi posed the conjecture that every minimal $k$ -factor-critical graph is of minimum degree $k + 1$ in 1998. In this paper, we confirm the conjecture for planar graphs.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .