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Über die Nichtexistenz zweier Knotenpunkte eines Graphen, die alle längsten Kreise fassen 这个中心是一个中心，每个人都拥有最长的圆圈，这个中心是不存在的
Journal of Combinatorial Theory Pub Date : 1970-04-01 DOI: 10.1016/S0021-9800(70)80085-0
Hansjoachim Walther
{"title":"Über die Nichtexistenz zweier Knotenpunkte eines Graphen, die alle längsten Kreise fassen","authors":"Hansjoachim Walther","doi":"10.1016/S0021-9800(70)80085-0","DOIUrl":"10.1016/S0021-9800(70)80085-0","url":null,"abstract":"<div><p>Es wird ein zweifach zusammenhängender (nichtplanarer) Graph angegeben, der keine zwei Knotenpunkte besitzt, so daß jeder längste Kreis des Graphen durch weinigstens einen der beiden Knotenpunkte geht.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 3","pages":"Pages 330-333"},"PeriodicalIF":0.0,"publicationDate":"1970-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80085-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86077497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
A combinatorial proof of Tucker's lemma for the n-cube n立方的塔克引理的组合证明
Journal of Combinatorial Theory Pub Date : 1970-04-01 DOI: 10.1016/S0021-9800(70)80081-3
James K. Baker
{"title":"A combinatorial proof of Tucker's lemma for the n-cube","authors":"James K. Baker","doi":"10.1016/S0021-9800(70)80081-3","DOIUrl":"10.1016/S0021-9800(70)80081-3","url":null,"abstract":"<div><p>Tucker's lemma is a combinatorial result which may be used to derive several theorems in topology. Some basic properties are established for the cube of integer lattice points. Tucker's lemma is then proved by applying a result which was originally presented for the octahedral subdivision of the <em>n</em>-disk.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 3","pages":"Pages 279-290"},"PeriodicalIF":0.0,"publicationDate":"1970-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80081-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88489534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Higher-Dimensional analogs of the four-color problem and some inequalities for simplicial complexes 四色问题的高维类比及简单复合体的一些不等式
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80071-0
Branko Grünbaum
{"title":"Higher-Dimensional analogs of the four-color problem and some inequalities for simplicial complexes","authors":"Branko Grünbaum","doi":"10.1016/S0021-9800(70)80071-0","DOIUrl":"10.1016/S0021-9800(70)80071-0","url":null,"abstract":"<div><p>Abstract</p><p>The four-color problem concerning planar graphs is shown to have meaningful higher-dimensional analogs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 147-153"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80071-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91156195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 18
The heawood map-coloring problem—Cases 1, 7, and 10 heawood地图上色问题——案例1、7和10
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80076-X
J.W.T. Youngs
{"title":"The heawood map-coloring problem—Cases 1, 7, and 10","authors":"J.W.T. Youngs","doi":"10.1016/S0021-9800(70)80076-X","DOIUrl":"10.1016/S0021-9800(70)80076-X","url":null,"abstract":"<div><p>This paper gives a proof of the fact that the chromatic number of an orientable surface of genus <em>p</em> is equal to the integral part of <span><math><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mn>48</mn><mi>p</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow></math></span> whenever the latter is congruent to 1, 7 or 10 modulo 12.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 220-231"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80076-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85550555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
The achromatic number of a graph 图的消色差数
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80072-2
Frnak Harary , Stephen Hedetniemi
{"title":"The achromatic number of a graph","authors":"Frnak Harary ,&nbsp;Stephen Hedetniemi","doi":"10.1016/S0021-9800(70)80072-2","DOIUrl":"10.1016/S0021-9800(70)80072-2","url":null,"abstract":"<div><p>The concept of coloring a graph has been shown to be subsumed by that of an homomorphism. This led in [3] to the definition of a complete <em>n</em>-coloring of a graph <em>G</em> and suggested therefore a new invariant, which we now call the “achromatic number” <em>ψ(G)</em>. While the chromatic number <em>χ(G)</em> is the minimum number of colors required for (a complete coloring of) the points of <em>G</em>, the achromatic number is the maximum such number. We obtain several bounds for <em>ψ(G)</em> in terms of other invariants of a graph, and in particular we show that, for any graph <em>G</em> having <em>p</em> points, <em><sub>x</sub>(G)+ͨ(G)¯⩽p+1</em>, a result which generalizes a theorem of Nordhaus and Gaddum [4].</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 154-161"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80072-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79268225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 101
Solution of the heawood map-coloring problem—Case 4 heawood地图上色问题的求解——案例4
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80074-6
C.M. Terry, L.R. Welch, J.W.T. Youngs
{"title":"Solution of the heawood map-coloring problem—Case 4","authors":"C.M. Terry,&nbsp;L.R. Welch,&nbsp;J.W.T. Youngs","doi":"10.1016/S0021-9800(70)80074-6","DOIUrl":"10.1016/S0021-9800(70)80074-6","url":null,"abstract":"<div><p>This paper gives a proof of the fact that the chromatic number of an orientable surface of genus <em>p</em> is equal to the integral part of <span><math><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mn>48</mn><mi>p</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow></math></span> whenever the latter is congruent to 4 modulo 12.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 170-174"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80074-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86329173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Solution of the heawood map-coloring problem—Cases 3, 5, 6, and 9 heawood地图上色问题的解法——案例3、5、6和9
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80075-8
J.W.T. Youngs
{"title":"Solution of the heawood map-coloring problem—Cases 3, 5, 6, and 9","authors":"J.W.T. Youngs","doi":"10.1016/S0021-9800(70)80075-8","DOIUrl":"10.1016/S0021-9800(70)80075-8","url":null,"abstract":"<div><p>This paper gives a proof of the fact that the chromatic number of an orientable surface of genus <em>p</em> is equal to the integral part of <span><math><mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mn>48</mn><mi>p</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow></mrow></math></span> whenever the latter is congruent to 3, 5, 6, or 9 modulo 12.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 175-219"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80075-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85623308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 27
A procedure for dissecting a rectangle into squares, and an example for the rectangle whose sides are in the ratio 2:1 将矩形分割成正方形的方法，以及边长比例为2:1的矩形的例子
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80077-1
R.L. Brooks
{"title":"A procedure for dissecting a rectangle into squares, and an example for the rectangle whose sides are in the ratio 2:1","authors":"R.L. Brooks","doi":"10.1016/S0021-9800(70)80077-1","DOIUrl":"https://doi.org/10.1016/S0021-9800(70)80077-1","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 232-243"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80077-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91683221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Operator theoretic invariants and the enumeration theory of Pólya and de Bruijn 算子论不变量与Pólya和de Bruijn的枚举理论
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80073-4
S.G. Williamson
{"title":"Operator theoretic invariants and the enumeration theory of Pólya and de Bruijn","authors":"S.G. Williamson","doi":"10.1016/S0021-9800(70)80073-4","DOIUrl":"10.1016/S0021-9800(70)80073-4","url":null,"abstract":"<div><p>It has been shown by M. Marcus and others that, in regard to combinatorial matrix functions and combinatorial inequalities, it is frequently fruitful to pass immediately from the consideration of permutations to the consideration of their tensor representations. Such an approach embeds the combinatorial arguments into the framework of linear algebra and frequently results in deeper theorems. It is interesting to note that certain basic combinatorial identities concerned with pattern enumeration and combinatorial generating functions can also be put into this framework. In this paper we consider one possible way of doing this.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 162-169"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80073-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88572263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Some simple perfect 2×1 rectangles 一些简单完美的2×1矩形
Journal of Combinatorial Theory Pub Date : 1970-03-01 DOI: 10.1016/S0021-9800(70)80078-3
P.J. Federico
{"title":"Some simple perfect 2×1 rectangles","authors":"P.J. Federico","doi":"10.1016/S0021-9800(70)80078-3","DOIUrl":"10.1016/S0021-9800(70)80078-3","url":null,"abstract":"","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 2","pages":"Pages 244-246"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80078-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88209003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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