关于没有诱导p5或k5−e的图

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2025-04-09 DOI:10.1002/jgt.23240

Arnab Char, T. Karthick

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Karthick","doi":"10.1002/jgt.23240","DOIUrl":null,"url":null,"abstract":"<div>\n \n In this paper, we are interested in some problems related to chromatic number and clique number for the class of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math>-free graphs and prove the following the results: (a) If <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> is a connected (<math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math>)-free graph with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>7</mn>\n </mrow>\n </mrow>\n </semantics></math>, then either <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> is the complement of a bipartite graph or <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> has a clique cut-set. Moreover, there is a connected (<math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math>)-free imperfect graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mn>6</mn>\n </mrow>\n </mrow>\n </semantics></math> and has no clique cut-set. This strengthens a result of Malyshev and Lobanova (Discrete Applied Mathematics 219 [2017] 158–166). (b) If <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> is a (<math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math>)-free graph with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>4</mn>\n </mrow>\n </mrow>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mi>max</mi>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>7</mn>\n \n <mo>,</mo>\n \n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math>. Moreover, the bound is tight when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∉</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mn>6</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math>. This result, together with known results, partially answers a question of Ju and Huang (Theoretical Computer Science 993 [2024] Article No.: 114465) and also improves a result of Xu [Manuscript 2022]. While Chromatic Number is known to be <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>N</mi>\n \n <mi>P</mi>\n </mrow>\n </mrow>\n </semantics></math>-hard for the class of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n </mrow>\n </mrow>\n </semantics></math>-free graphs, our results, together with some known results, imply that Chromatic Number can be solved in polynomial time for the class of (<math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>P</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>e</mi>\n </mrow>\n </mrow>\n </semantics></math>)-free graphs, which may be of independent interest.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"5-22"},"PeriodicalIF":1.0000,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Graphs With No Induced \\n \\n \\n \\n \\n P\\n 5\\n \\n \\n \\n or \\n \\n \\n \\n \\n K\\n 5\\n \\n −\\n e\",\"authors\":\"Arnab Char, T. Karthick\",\"doi\":\"10.1002/jgt.23240\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n In this paper, we are interested in some problems related to chromatic number and clique number for the class of <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <msub>\\n <mi>P</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mi>e</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math>-free graphs and prove the following the results: (a) If <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> is a connected (<math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>P</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mi>e</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>)-free graph with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ω</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mn>7</mn>\\n </mrow>\\n </mrow>\\n </semantics></math>, then either <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> is the complement of a bipartite graph or <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> has a clique cut-set. Moreover, there is a connected (<math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>P</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mi>e</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>)-free imperfect graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ω</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>H</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>=</mo>\\n \\n <mn>6</mn>\\n </mrow>\\n </mrow>\\n </semantics></math> and has no clique cut-set. This strengthens a result of Malyshev and Lobanova (Discrete Applied Mathematics 219 [2017] 158–166). (b) If <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> is a (<math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>P</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mi>e</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>)-free graph with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ω</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n </mrow>\\n </semantics></math>, then <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>χ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≤</mo>\\n \\n <mi>max</mi>\\n \\n <mrow>\\n <mo>{</mo>\\n \\n <mrow>\\n <mn>7</mn>\\n \\n <mo>,</mo>\\n \\n <mi>ω</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math>. Moreover, the bound is tight when <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ω</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∉</mo>\\n \\n <mrow>\\n <mo>{</mo>\\n \\n <mrow>\\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>5</mn>\\n \\n <mo>,</mo>\\n \\n <mn>6</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math>. This result, together with known results, partially answers a question of Ju and Huang (Theoretical Computer Science 993 [2024] Article No.: 114465) and also improves a result of Xu [Manuscript 2022]. 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引用次数: 0

摘要

本文研究了（p5）类的色数和团数的相关问题。k5−e)自由图，并证明了以下结果：(a)如果G是连通的（p5），与ω (G)的k5−e自由图)≥7；那么要么G是二部图的补，要么G有团切集。此外，还有一个连接的（p5），k5−e)自由不完美图H withω (H) = 6，无团切集。这加强了Malyshev和Lobanova （Discrete Applied Mathematics 219[2017] 158-166）的结果。 (b)若G为a (p5)，与ω (G)的k5−e自由图)≥4；则χ (G)≤max {7, ω (g)}。此外,当ω (G)∈{4， 5,6}。这一结果，连同已知的结果，部分地回答了菊和黄（理论计算机科学993[2024]第2篇）提出的一个问题。： 114465)，也改进了Xu [Manuscript 2022]的结果。虽然已知色数对于p5自由图来说是np -困难的，我们的结果是，结合一些已知的结果，表明对于（p5）类的色数可以在多项式时间内求解。k5−e)自由图，这可能是独立的兴趣。

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On Graphs With No Induced P 5 or K 5 − e

In this paper, we are interested in some problems related to chromatic number and clique number for the class of $(P_{5}, K_{5} - e)$ -free graphs and prove the following the results: (a) If $G$ is a connected ( $P_{5}, K_{5} - e$ )-free graph with $ω (G) \geq 7$ , then either $G$ is the complement of a bipartite graph or $G$ has a clique cut-set. Moreover, there is a connected ( $P_{5}, K_{5} - e$ )-free imperfect graph $H$ with $ω (H) = 6$ and has no clique cut-set. This strengthens a result of Malyshev and Lobanova (Discrete Applied Mathematics 219 [2017] 158–166). (b) If $G$ is a ( $P_{5}, K_{5} - e$ )-free graph with $ω (G) \geq 4$ , then $χ (G) \leq \max {7, ω (G)}$ . Moreover, the bound is tight when $ω (G) \notin {4, 5, 6}$ . This result, together with known results, partially answers a question of Ju and Huang (Theoretical Computer Science 993 [2024] Article No.: 114465) and also improves a result of Xu [Manuscript 2022]. While Chromatic Number is known to be $N P$ -hard for the class of $P_{5}$ -free graphs, our results, together with some known results, imply that Chromatic Number can be solved in polynomial time for the class of ( $P_{5}, K_{5} - e$ )-free graphs, which may be of independent interest.

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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 .