Debra L. Boutin, S. Cockburn, L. Keough, Sarah Loeb, Puck Rombach
{"title":"Symmetry parameters of various hypercube families","authors":"Debra L. Boutin, S. Cockburn, L. Keough, Sarah Loeb, Puck Rombach","doi":"10.26493/2590-9770.1481.29d","DOIUrl":"https://doi.org/10.26493/2590-9770.1481.29d","url":null,"abstract":"In this paper we study the symmetry parameters determining number, distinguishing number, and cost of 2-distinguishing, for some variations on hypercubes, namely Hamming graphs, powers of hypercubes, folded hypercubes, enhanced hypercubes, augmented hypercubes and locally twisted hypercubes.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114977713","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}
{"title":"On algebraic structure of the Reed-Muller codes","authors":"Harinaivo Andriatahiny","doi":"10.26493/2590-9770.1417.02a","DOIUrl":"https://doi.org/10.26493/2590-9770.1417.02a","url":null,"abstract":"It is known that the Reed-Muller codes over a prime field may be described as the radical powers of a modular group algebra. In this paper, we give a new proof of the same result in a quotient of a polynomial ring. Special elements in a prime field are studied. An interpolation polynomial is introduced in order to characterize the coefficients of the Jennings polynomials.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"193 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123107785","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}
{"title":"Constructing integer-magic graphs via the Combinatorial Nullstellensatz","authors":"R. Low, D. Roberts","doi":"10.26493/2590-9770.1401.a6a","DOIUrl":"https://doi.org/10.26493/2590-9770.1401.a6a","url":null,"abstract":"Let A be a nontrivial abelian group and A* = A {0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to construct nontrivial classes of ℤp-magic graphs, prime p ≥ 3. For these graphs, some lower bounds on the number of distinct ℤp-magic labelings are also established.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129328413","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}
{"title":"Self-dual polyhedra of given degree sequence","authors":"Riccardo W. Maffucci","doi":"10.26493/2590-9770.1537.cf9","DOIUrl":"https://doi.org/10.26493/2590-9770.1537.cf9","url":null,"abstract":"Given vertex valencies admissible for a self-dual polyhedral graph, we describe an algorithm to explicitly construct such a polyhedron. Inputting in the algorithm permutations of the degree sequence can give rise to non-isomorphic graphs. As an application, we find as a function of $ngeq 3$ the minimal number of vertices for a self-dual polyhedron with at least one vertex of degree $i$ for each $3leq ileq n$, and construct such polyhedra. Moreover, we find a construction for non-self-dual polyhedral graphs of minimal order with at least one vertex of degree $i$ and at least one $i$-gonal face for each $3leq ileq n$.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113960473","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}
{"title":"On hamiltonian cycles in Cayley graphs of order pqrs","authors":"D. Morris","doi":"10.26493/2590-9770.1442.cb1","DOIUrl":"https://doi.org/10.26493/2590-9770.1442.cb1","url":null,"abstract":"Let $G$ be a finite group. We show that if $|G| = pqrs$, where $p$, $q$, $r$, and $s$ are distinct odd primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121267492","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}
{"title":"On avoiding 1233","authors":"T. Mansour, M. Shattuck","doi":"10.26493/2590-9770.1377.8e9","DOIUrl":"https://doi.org/10.26493/2590-9770.1377.8e9","url":null,"abstract":"In this paper, we establish a recurrence relation for finding the generating function for the number of k-ary words of length n that avoid 1233 for arbitrary k. Comparable generating function formulas may also be found counting words where a single permutation pattern of length three is avoided in addition to 1233.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"290 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116241584","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}
{"title":"The string C-group representations of the Suzuki, Rudvalis and O'Nan sporadic groups","authors":"D. Leemans, J. Mulpas","doi":"10.26493/2590-9770.1405.4ce","DOIUrl":"https://doi.org/10.26493/2590-9770.1405.4ce","url":null,"abstract":"We present new algorithms to classify all string C-group representations of a given group G. We use these algorithms to classify all string C-group representations of the Suzuki, Rudvalis and O’Nan sporadic groups. The new rank three algorithms also permit us to get all string C-group representations of rank three for the Conway group Co2 and the Fischer group Fi22.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124616412","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}
{"title":"Cayley graphs of order 6pq and 7pq are Hamiltonian","authors":"Farzad Maghsoudi","doi":"10.26493/2590-9770.1389.fa2","DOIUrl":"https://doi.org/10.26493/2590-9770.1389.fa2","url":null,"abstract":"Assume G is a finite group, such that |G| = 6pq or 7pq, where p and q are distinct prime numbers, and let S be a generating set of G. We prove there is a Hamiltonian cycle in the corresponding Cayley graph Cay(G;S).","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127139852","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}
{"title":"Sierpiński products of r-uniform hypergraphs","authors":"Mark Budden, Josh Hiller","doi":"10.26493/2590-9770.1402.D50","DOIUrl":"https://doi.org/10.26493/2590-9770.1402.D50","url":null,"abstract":"If H1 and H2 are r-uniform hypergraphs and f is a function from the set of all (r − 1)-element subsets of V(H1) into V(H2), then the Sierpinski product H1⊗fH2 is defined to have vertex set V(H1) × V(H2) and hyperedges falling into two classes: (g, h1)(g, h2)⋯(g, hr), such that g ∈ V(H1) and h1h2⋯hr ∈ E(H2),and (g1, f({g2, g3, …, gr}))(g2, f({g1, g3, …, gr}))⋯(gr, f({g1, g2, …, gr − 1})),such that g1g2⋯gr ∈ E(H1). We develop the basic structure possessed by this product and offer proofs of numerous extremal properties involving connectivity, clique numbers, and strong chromatic numbers.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121297757","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}
{"title":"Paint cost and the frugal distinguishing number","authors":"Debra L. Boutin","doi":"10.26493/2590-9770.1463.f59","DOIUrl":"https://doi.org/10.26493/2590-9770.1463.f59","url":null,"abstract":"You are handed a graph with vertices in a neutral color and asked to color a subset of vertices with expensive paints in $d$ colors in such a way that only the trivial symmetry preserves the color classes. Your goal is to minimize the number of vertices needing this expensive paint. This paper address the issues surrounding your choices. In particular, a graph is said to be $d$-distinguishable if there exists a coloring with $d$ colors so that only the trivial automorphism preserves the color classes. The distinguishing number of $G$, denoted ${rm Dist}(G)$, is the smallest $d$ for which $G$ is $d$-distinguishable. We define the -paint cost of $d$-distinguishing, denoted $rho^d(G)$, to be the minimum number of vertices that need to be painted to $d$-distinguish $G$. This cost varies with $d$. The maximum paint cost for $G$ is called the upper paint cost, denoted $rho^u(G)$, and occurs when $d={rm Dist}(G)$; the minimum paint cost is called the lower paint cost, denoted $rho^ell(G)$. Further, we define the smallest $d$ for which the paint cost is $rho^ell(G)$, to be the frugal distinguishing number, ${rm Fdist}(G)$. In this paper we formally define $rho^d(G)$, $rho^u(G)$, $rho^ell(G)$, and ${rm Fdist}(G)$. We also show that $rho^u(G)$ and $rho^ell(G)$, as well as ${rm Fdist}(G)$ and ${rm Dist}(G)$, can be arbitrarily large multiples of each other. Lastly, we find these parameters for the book graph $B_{m,n}$, summarized as follows. For $ngeq 2$ and $mgeq 4$, we show $bullet$ $rho^ell(B_{m,n}) = n-1;$ $bullet$ $rho^u(B_{m,n}) geq (m-2) left( n-k^{m-3} right) +1$, where $k={rm Dist}(B_{m,n});$ $bullet$ ${rm Fdist}(B_{m,n}) = 2+leftlfloor frac{n-1}{m-2} rightrfloor.$","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124635317","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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