关于强化Kundu k因子定理的一个猜想

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":"关于强化Kundu k因子定理的一个猜想","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":"{\"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}","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

摘要

设π = （d1）... ,d) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0002”威利:位置= "方程/ jgt23177 -数学- 0002. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> \ unicode {x003C0} & lt; / mi> & lt; mo> = & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mo> \ unicode {x02026} & lt; / mo> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>是偶数n&lt； 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></mrow></ mrow></mrow></ mrow></mrow></ mrow></math&gt；．在1974年,Kundu证明，如果dk (π) = (d1−k，…，d n−k) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0004 .png"><mrow>< msub><mi class=“ mjx - text - caligrapic ”mathvariant = "脚本"祝辞D< / mi> & lt; mi> k< / mi> & lt; / msub> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt \ unicode {x003C0} & lt; / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo> = & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; msub> & lt; mi> D< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mi> k< / mi> & lt; mo>, & lt; / mo> & lt; mo> \ unicode {x02026} & lt; / mo> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> D< / mi> & lt; mi> nlt; / mi> & lt; / msub> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mi> k< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>是图形化的，那么一些实现π &lt；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>< \unicode{x003C0}</ mrow></mrow></ mrow></mrow></ mrow></math&gt；有一个k &lt；数学xmlns="http://www.w3。 org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0006 .png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；因素。对于r≤2 &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0007”威利:位置= "方程/ jgt23177 -数学- 0007. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>,Busch等人及后来的Seacrest for r≤4 &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0008”威利:位置= "方程/ jgt23177 -数学- 0008. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mn> 4 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>显示，如果r≤k &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0009”威利:位置= "方程/ jgt23177 -数学- 0009. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mi> k< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>和k (π) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0010 .png"><mrow><mrow>< mrow><mrow>< mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0010.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0010.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ mrow></ mjx - text - caligrapic " mathvariant="script">D</ mrow>&lt；如果是图形化的，则可以使用k&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0011 .png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></math&gt；-因子，其边可以划分为（k−r） &ltxmlns = " http://www.w3.org/1998/Math/MathML " altimg = " urn: x-wiley: 03649024:媒体:jgt23177: jgt23177 -数学- 0012“威利:位置=“方程/ jgt23177 -数学- 0012. - png”祝辞& lt; mrow> & lt; mrow> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> k< / mi> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mi> r< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>-factor and r< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0013 .png">< <mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；edge-disjoint为1的因子。我们把它改进成任意r≤imn {k+ 5 3，k} <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0014”威利:位置= "方程/ jgt23177 -数学- 0014. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mi> m< / mi> & lt; mi> i< / mi> & lt; mi> n< / mi> & lt; mrow> & lt;莫弹性=“true”在{& lt; / mo> & lt; mfenced近= " \ unicode {x02309} "打开= " \ unicode {x02308} "祝辞& lt; mfrac> & lt; mrow> & lt; mi> k< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 5 & lt; / mn> & lt; / mrow> & lt; mn> 3 & lt; / mn> & lt; / mfrac> & lt; / mfenced> & lt; mo> & lt; / mo> & lt; mi> k< / mi> & lt;莫弹性=“true”祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>． 1978年Brualdi和2012年Busch等人，推测r = k <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0015”威利:位置= "方程/ jgt23177 -数学- 0015. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mo> = & lt; / mo> & lt; mi> k< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>．对于k≥6的猜想仍然是开放的&lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0016”威利:位置= "方程/ jgt23177 -数学- 0016. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> k< / mi> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mn> 6 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>．然而,Busch等人证明，当d1≤n2 + 1时，猜想成立<math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0017”威利:位置= "方程/ jgt23177 -数学- 0017. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mfrac> & lt; mi> n< / mi> & lt; mn> 2 & lt; / mn> & lt; / mfrac> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>或者d n≥n 2 + k−2 <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”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mfrac> & lt; mi> n< / mi> & lt; mn> 2 & lt; / mn> & lt; / mfrac> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mi> k< / mi> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>．我们首先通过开发推广边缘交换的新工具来探索这一猜想。有了这些新工具，我们可以放弃假设D k (π) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg="urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0019 .png"><mrow><mrow>< mrow><mrow>< mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0019.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0019.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ mrow></ mjx - text - caligrapic " mathvariant="script">D</ mrow>&lt；如果d d 1−dN + k≥d1−D n + k−1，<math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0020”威利:位置= "方程/ jgt23177 -数学- 0020. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mi> k< / mi> & lt; / mrow> & lt; / msub> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mi> k< / mi> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; mo> & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / math>然后π &lt；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>< \unicode{x003C0}</ mrow></mrow></ mrow></mrow>&lt；有一个实现k&lt； 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></mrow></ mrow></mrow></ mrow></mrow></math&gt；edge-disjoint为1的因子。由此我们证实了n≥d1时的猜想+ k−1 2 <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0023”威利:位置= "方程/ jgt23177 -数学- 0023. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mfrac> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mi> k< / mi> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; mn> 2 & lt; / mn> & lt; / mfrac> & lt; / mrow> & lt; / mrow> & lt; / math>或者D k (π) <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>< mrow><mrow>< mrow><mrow>< mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0024.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ jgt23177-math-0024.png"><mrow>< / mjx - text - caligrapic " mathvariant="script">D</ mrow></ mjx - text - caligrapic " mathvariant="script">D</ mrow>< (</mo>< /msub><mrow>&lt；是图形和d1≤ma x {n∕2 + d n−k；（n + d n）∕2} <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23177:jgt23177-math-0025”威利:位置= "方程/ jgt23177 -数学- 0025. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> d< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> \ unicode {x02264} & lt; / mo> & lt; mi> m< / mi> & lt; mi> a< / mi> & lt; mi> x< / mi> & lt;莫弹性= " false "祝辞{& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo> \ unicode {x02215} & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo> \ unicode {x02212} & lt; / mo> & lt; mi> k< / mi> & lt; mo>, & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt n< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; msub> & lt; mi> d< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; mo>) & lt; / mo> & lt;/ mrow> & lt; mo&gt \ unicode {x02215} & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; / mrow> & lt;莫弹性=“false”祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / math>．

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On a conjecture that strengthens Kundu's k

k

-factor theorem

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 .