Pub Date : 2023-01-04DOI: 10.26493/2590-9770.1516.96f
T. Mansour
{"title":"The length of the initial longest increasing sequence in a permutation","authors":"T. Mansour","doi":"10.26493/2590-9770.1516.96f","DOIUrl":"https://doi.org/10.26493/2590-9770.1516.96f","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43258266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-22DOI: 10.26493/2590-9770.1540.7b1
M. Staš, M. Švecová
{"title":"Disconnected spanning subgraphs of paths in the join products with cycles","authors":"M. Staš, M. Švecová","doi":"10.26493/2590-9770.1540.7b1","DOIUrl":"https://doi.org/10.26493/2590-9770.1540.7b1","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48549468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-24DOI: 10.26493/2590-9770.1395.a37
W. Imrich, T. Lachmann, T. Tucker, G. Wiegel
A set S of vertices in a graph G with nontrivial automorphism group is asymmetrizing if the identity mapping is the only automorphism of G that preserves S as a set. If such sets exist, then their minimum cardinality is the asymmetrizing cost ρ ( G ) of G . For finite graphs the asymmetrizing density δ ( G ) of G is the quotient of the size of S by the order of G . For infinite graphs δ ( G ) is defined by a limit process. Many classes of infinite graphs with δ ( G ) = 0 are known, but seemingly no infinite vertex transitive graphs with δ ( G ) > 0. Here, we construct connected, infinite vertex transitive cubic graphs of asymmetrizing density δ ( G ) = 1 n 2 n +1 for each n ≥ 1. We also construct finite vertex transitive cubic graphs of arbitrarily large asymmetrizing cost. The examples are Split Praeger–Xu graphs, for which we provide another characterization. This contrasts with our results for vertex transitive cubic graphs that have one arc orbit or are so-called synchronously connected graphs with two arc orbits. For them we show that ρ ( G ) is either ≤ 5 or infinite. In the latter case δ ( G ) = 0.
{"title":"Asymmetrizing cost and density of vertex-transitive cubic graphs","authors":"W. Imrich, T. Lachmann, T. Tucker, G. Wiegel","doi":"10.26493/2590-9770.1395.a37","DOIUrl":"https://doi.org/10.26493/2590-9770.1395.a37","url":null,"abstract":"A set S of vertices in a graph G with nontrivial automorphism group is asymmetrizing if the identity mapping is the only automorphism of G that preserves S as a set. If such sets exist, then their minimum cardinality is the asymmetrizing cost ρ ( G ) of G . For finite graphs the asymmetrizing density δ ( G ) of G is the quotient of the size of S by the order of G . For infinite graphs δ ( G ) is defined by a limit process. Many classes of infinite graphs with δ ( G ) = 0 are known, but seemingly no infinite vertex transitive graphs with δ ( G ) > 0. Here, we construct connected, infinite vertex transitive cubic graphs of asymmetrizing density δ ( G ) = 1 n 2 n +1 for each n ≥ 1. We also construct finite vertex transitive cubic graphs of arbitrarily large asymmetrizing cost. The examples are Split Praeger–Xu graphs, for which we provide another characterization. This contrasts with our results for vertex transitive cubic graphs that have one arc orbit or are so-called synchronously connected graphs with two arc orbits. For them we show that ρ ( G ) is either ≤ 5 or infinite. In the latter case δ ( G ) = 0.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48043727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-03DOI: 10.26493/2590-9770.1555.9a6
M. Horňák
A vertex colouring $f:V(G)to C$ of a graph $G$ is complete if for any $c_1,c_2in C$ with $c_1ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number $mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. Let $G_1square G_2$ denote the Cartesian product of graphs $G_1$ and $G_2$. In the paper $mathrm{achr}(K_{r^2+r+1}square K_q)$ is determined for an infinite number of $q$s provided that $r$ is a finite projective plane order.
{"title":"On the achromatic number of the Cartesian product of two complete graphs","authors":"M. Horňák","doi":"10.26493/2590-9770.1555.9a6","DOIUrl":"https://doi.org/10.26493/2590-9770.1555.9a6","url":null,"abstract":"A vertex colouring $f:V(G)to C$ of a graph $G$ is complete if for any $c_1,c_2in C$ with $c_1ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number $mathrm{achr}(G)$ of colours in a proper complete vertex colouring of $G$. Let $G_1square G_2$ denote the Cartesian product of graphs $G_1$ and $G_2$. In the paper $mathrm{achr}(K_{r^2+r+1}square K_q)$ is determined for an infinite number of $q$s provided that $r$ is a finite projective plane order.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"25 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2022-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69338932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-21DOI: 10.26493/2590-9770.1468.a2e
Shao-Fei Du, Kai Yuan
{"title":"On polynomials of small degree over finite fields representing quadratic residues","authors":"Shao-Fei Du, Kai Yuan","doi":"10.26493/2590-9770.1468.a2e","DOIUrl":"https://doi.org/10.26493/2590-9770.1468.a2e","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47893082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-04DOI: 10.26493/2590-9770.1510.22d
P. Fowler, Wendy J. Myrvold, Rebecca L. Vandenberg, E. Hartung, J. Graver
{"title":"Clar and Fries structures for fullerenes","authors":"P. Fowler, Wendy J. Myrvold, Rebecca L. Vandenberg, E. Hartung, J. Graver","doi":"10.26493/2590-9770.1510.22d","DOIUrl":"https://doi.org/10.26493/2590-9770.1510.22d","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42993345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.26493/2590-9770.1473.ec5
Lorenzo Sauras Altuzarra
{"title":"Some properties of the factors of Fermat numbers","authors":"Lorenzo Sauras Altuzarra","doi":"10.26493/2590-9770.1473.ec5","DOIUrl":"https://doi.org/10.26493/2590-9770.1473.ec5","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"6 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69338925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-05DOI: 10.26493/2590-9770.1420.B57
Yaokun Wu, Da Zhao
Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrodinger operators on countable digraphs. We suggest the open question of deciding if the countability assumption can be dropped from our main results.
{"title":"Three conjectures of Ostrander on digraph Laplacian eigenvectors","authors":"Yaokun Wu, Da Zhao","doi":"10.26493/2590-9770.1420.B57","DOIUrl":"https://doi.org/10.26493/2590-9770.1420.B57","url":null,"abstract":"Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrodinger operators on countable digraphs. We suggest the open question of deciding if the countability assumption can be dropped from our main results.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41976090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-29DOI: 10.26493/2590-9770.1408.f90
G. Gévay, Leah Wrenn Berman, T. Pisanski
In a series of papers and in his 2009 book on configurations Branko Grunbaum described a sequence of operations to produce new (n4) configurations from various input configurations. These operations were later called the “Grunbaum Incidence Calculus”. We generalize two of these operations to produce operations on arbitrary (nk) configurations. Using them, we show that for any k there exists an integer Nk such that for any n ≥ Nk there exists a geometric (nk) configuration. We use empirical results for k = 2, 3, 4, and some more detailed analysis to improve the upper bound for larger values of k.
{"title":"Connected geometric (n_k) configurations exist for almost all n","authors":"G. Gévay, Leah Wrenn Berman, T. Pisanski","doi":"10.26493/2590-9770.1408.f90","DOIUrl":"https://doi.org/10.26493/2590-9770.1408.f90","url":null,"abstract":"In a series of papers and in his 2009 book on configurations Branko Grunbaum described a sequence of operations to produce new (n4) configurations from various input configurations. These operations were later called the “Grunbaum Incidence Calculus”. We generalize two of these operations to produce operations on arbitrary (nk) configurations. Using them, we show that for any k there exists an integer Nk such that for any n ≥ Nk there exists a geometric (nk) configuration. We use empirical results for k = 2, 3, 4, and some more detailed analysis to improve the upper bound for larger values of k.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42469045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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