On a conjecture that strengthens Kundu's k $k$ -factor theorem

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-10-06 DOI:10.1002/jgt.23177

James M. Shook

{"title":"On a conjecture that strengthens Kundu's \n \n \n \n k\n \n \n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0001\" wiley:location=\"equation/jgt23177-math-0001.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math>\n -factor theorem","authors":"James M. Shook","doi":"10.1002/jgt.23177","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>π</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0002\" wiley:location=\"equation/jgt23177-math-0002.png\"><mrow><mrow><mi>\\unicode{x003C0}</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>,</mo><mo>\\unicode{x02026}</mo><mo>,</mo><msub><mi>d</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> be a nonincreasing degree sequence with even <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0003\" wiley:location=\"equation/jgt23177-math-0003.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>. In 1974, Kundu showed that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>D</mi>\n \n <mi>k</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>π</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>−</mo>\n \n <mi>k</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0004\" wiley:location=\"equation/jgt23177-math-0004.png\"><mrow><mrow><msub><mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>\\unicode{x003C0}</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x02212}</mo><mi>k</mi><mo>,</mo><mo>\\unicode{x02026}</mo><mo>,</mo><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x02212}</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is graphic, then some realization of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>π</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0005\" wiley:location=\"equation/jgt23177-math-0005.png\"><mrow><mrow><mi>\\unicode{x003C0}</mi></mrow></mrow></math></annotation>\n </semantics></math> has a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0006\" wiley:location=\"equation/jgt23177-math-0006.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-factor. For <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0007\" wiley:location=\"equation/jgt23177-math-0007.png\"><mrow><mrow><mi>r</mi><mo>\\unicode{x02264}</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>, Busch et al. and later Seacrest for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mo>≤</mo>\n \n <mn>4</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0008\" wiley:location=\"equation/jgt23177-math-0008.png\"><mrow><mrow><mi>r</mi><mo>\\unicode{x02264}</mo><mn>4</mn></mrow></mrow></math></annotation>\n </semantics></math> showed that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mo>≤</mo>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0009\" wiley:location=\"equation/jgt23177-math-0009.png\"><mrow><mrow><mi>r</mi><mo>\\unicode{x02264}</mo><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>D</mi>\n \n <mi>k</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>π</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0010\" wiley:location=\"equation/jgt23177-math-0010.png\"><mrow><mrow><msub><mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>\\unicode{x003C0}</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is graphic, then there is a realization with a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0011\" wiley:location=\"equation/jgt23177-math-0011.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-factor whose edges can be partitioned into a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mi>r</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0012\" wiley:location=\"equation/jgt23177-math-0012.png\"><mrow><mrow><mrow><mo>(</mo><mrow><mi>k</mi><mo>\\unicode{x02212}</mo><mi>r</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>-factor and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0013\" wiley:location=\"equation/jgt23177-math-0013.png\"><mrow><mrow><mi>r</mi></mrow></mrow></math></annotation>\n </semantics></math> edge-disjoint 1-factors. We improve this to any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mo>≤</mo>\n \n <mi>m</mi>\n \n <mi>i</mi>\n \n <mi>n</mi>\n \n <mrow>\n <mo>{</mo>\n \n <mfenced>\n <mfrac>\n <mrow>\n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>5</mn>\n </mrow>\n \n <mn>3</mn>\n </mfrac>\n </mfenced>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0014\" wiley:location=\"equation/jgt23177-math-0014.png\"><mrow><mrow><mi>r</mi><mo>\\unicode{x02264}</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo stretchy=\"true\">{</mo><mfenced close=\"\\unicode{x02309}\" open=\"\\unicode{x02308}\"><mfrac><mrow><mi>k</mi><mo>\\unicode{x0002B}</mo><mn>5</mn></mrow><mn>3</mn></mfrac></mfenced><mo>,</mo><mi>k</mi><mo stretchy=\"true\">}</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>. In 1978, Brualdi and then Busch et al. in 2012, conjectured that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mo>=</mo>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0015\" wiley:location=\"equation/jgt23177-math-0015.png\"><mrow><mrow><mi>r</mi><mo>=</mo><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>. The conjecture is still open for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>6</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0016\" wiley:location=\"equation/jgt23177-math-0016.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x02265}</mo><mn>6</mn></mrow></mrow></math></annotation>\n </semantics></math>. However, Busch et al. showed the conjecture is true when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>≤</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mn>2</mn>\n </mfrac>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0017\" wiley:location=\"equation/jgt23177-math-0017.png\"><mrow><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x02264}</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math> or <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>≥</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mn>2</mn>\n </mfrac>\n \n <mo>+</mo>\n \n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0018\" wiley:location=\"equation/jgt23177-math-0018.png\"><mrow><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x02265}</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>\\unicode{x0002B}</mo><mi>k</mi><mo>\\unicode{x02212}</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>. We explore this conjecture by first developing new tools that generalize edge-exchanges. With these new tools, we can drop the assumption <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>D</mi>\n \n <mi>k</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>π</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0019\" wiley:location=\"equation/jgt23177-math-0019.png\"><mrow><mrow><msub><mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>\\unicode{x003C0}</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is graphic and show that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>+</mo>\n \n <mi>k</mi>\n </mrow>\n </msub>\n \n <mo>≥</mo>\n \n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>+</mo>\n \n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0020\" wiley:location=\"equation/jgt23177-math-0020.png\"><mrow><mrow><msub><mi>d</mi><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x02212}</mo><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x0002B}</mo><mi>k</mi></mrow></msub><mo>\\unicode{x02265}</mo><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x02212}</mo><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x0002B}</mo><mi>k</mi><mo>\\unicode{x02212}</mo><mn>1</mn><mo>,</mo></mrow></mrow></math></annotation>\n </semantics></math> then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>π</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0021\" wiley:location=\"equation/jgt23177-math-0021.png\"><mrow><mrow><mi>\\unicode{x003C0}</mi></mrow></mrow></math></annotation>\n </semantics></math> has a realization with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0022\" wiley:location=\"equation/jgt23177-math-0022.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> edge-disjoint 1-factors. From this we confirm the conjecture when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>≥</mo>\n \n <mfrac>\n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>+</mo>\n \n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mn>2</mn>\n </mfrac>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0023\" wiley:location=\"equation/jgt23177-math-0023.png\"><mrow><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x02265}</mo><mfrac><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x0002B}</mo><mi>k</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></mrow></math></annotation>\n </semantics></math> or when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>D</mi>\n \n <mi>k</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>π</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0024\" wiley:location=\"equation/jgt23177-math-0024.png\"><mrow><mrow><msub><mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>\\unicode{x003C0}</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is graphic and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>d</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>≤</mo>\n \n <mi>m</mi>\n \n <mi>a</mi>\n \n <mi>x</mi>\n \n <mo>{</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>∕</mo>\n \n <mn>2</mn>\n \n <mo>+</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>−</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mi>n</mi>\n \n <mo>+</mo>\n \n <msub>\n <mi>d</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∕</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0025\" wiley:location=\"equation/jgt23177-math-0025.png\"><mrow><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>\\unicode{x02264}</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy=\"false\">{</mo><mrow><mi>n</mi><mo>\\unicode{x02215}</mo><mn>2</mn><mo>\\unicode{x0002B}</mo><msub><mi>d</mi><mi>n</mi></msub><mo>\\unicode{x02212}</mo><mi>k</mi><mo>,</mo><mrow><mo>(</mo><mi>n</mi><mo>\\unicode{x0002B}</mo><msub><mi>d</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>\\unicode{x02215}</mo><mn>2</mn></mrow><mo stretchy=\"false\">}</mo></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"463-491"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23177","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $π = (d_{1}, \dots, d_{n})$ be a nonincreasing degree sequence with even $n$ . In 1974, Kundu showed that if $D_{k} (π) = (d_{1} - k, \dots, d_{n} - k)$ is graphic, then some realization of $π$ has a $k$ -factor. For $r \leq 2$ , Busch et al. and later Seacrest for $r \leq 4$ showed that if $r \leq k$ and $D_{k} (π)$ is graphic, then there is a realization with a $k$ -factor whose edges can be partitioned into a $(k - r)$ -factor and $r$ edge-disjoint 1-factors. We improve this to any $r \leq m i n {(\frac{k + 5}{3}), k}$ . In 1978, Brualdi and then Busch et al. in 2012, conjectured that $r = k$ . The conjecture is still open for $k \geq 6$ . However, Busch et al. showed the conjecture is true when $d_{1} \leq \frac{n}{2} + 1$ or $d_{n} \geq \frac{n}{2} + k - 2$ . We explore this conjecture by first developing new tools that generalize edge-exchanges. With these new tools, we can drop the assumption $D_{k} (π)$ is graphic and show that if $d_{d_{1} - d_{n} + k} \geq d_{1} - d_{n} + k - 1,$ then $π$ has a realization with $k$ edge-disjoint 1-factors. From this we confirm the conjecture when $d_{n} \geq \frac{d_{1} + k - 1}{2}$ or when $D_{k} (π)$ is graphic and $d_{1} \leq m a x {n ∕ 2 + d_{n} - k, (n + d_{n}) ∕ 2}$ .

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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 .

On a conjecture that strengthens Kundu's k k -factor theorem

Abstract

On a conjecture that strengthens Kundu's k $k$ -factor theorem