Let D be any of the 10 digraphs obtained by orienting the edges of K 4 -e. We establish necessary... more Let D be any of the 10 digraphs obtained by orienting the edges of K 4 -e. We establish necessary and sufficient conditions for the existence of a (K * n , D)-design for 8 of these digraphs. Partial results as well as some nonexistence results are established for the remaining 2 digraphs.
For all integers m, nand t, we determine necessary and sufficient conditions for the existence of... more For all integers m, nand t, we determine necessary and sufficient conditions for the existence of (1) a pair of 3-cube decompositions of Kn having precisely t common 3-cubes; and (2) a pair of 3-cube decompositions of Km,n having precisely t common 3-cubes.
ABSTRACT We introduce the concept of a free α-valuation of a graph, and prove that the vertex-dis... more ABSTRACT We introduce the concept of a free α-valuation of a graph, and prove that the vertex-disjoint union of any collection of graphs with free α-valuations has an α-valuation. Many bipartite graphs have free α-valuations including the complete bipartite graph K m,n when m>1 and n>2, and the d-cube Q d for d>2.
A k-factorization of the complete t-uniform hypergraph v . For v ≤ 9, we use nauty to generate th... more A k-factorization of the complete t-uniform hypergraph v . For v ≤ 9, we use nauty to generate the 2-regular and 3-regular spanning subhypergraphs of K (3) v and investigate which of these subhypergraphs factorize v -I, where I is a 1-factor. We settle this question for all but two of these subhypergraphs.
We show via an exhaustive computer search that there does not exist a (K 6 -e)decomposition of K ... more We show via an exhaustive computer search that there does not exist a (K 6 -e)decomposition of K 29 . This is the first example of a non-complete graph G for which a G-decomposition of K 2|E(G)|+1 does not exist.
For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on th... more For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on the number of cube and bipartition vertices must hold. We prove these necessary conditions sufficient in some cases. For cubes of fixed dimension d (indeed for d-regular bipartite graphs in general) we show that proving sufficiency can be reduced to decomposing a finite number of complete bipartite graphs. When t = 2d-' and r is the remainder on dividing t by d, we show Kr,r is decomposable into d-cubes and an r-factor, where if r > 0 this r-factor itself is decomposable into r-cubes.
Information Systems Education Journal, Jun 1, 2018
Our nation's competitive edge is highly dependent on the success of STEM education and the abilit... more Our nation's competitive edge is highly dependent on the success of STEM education and the ability of information technology (IT) graduates to find jobs. The School of Information Technology at Illinois State University (ISU) is strategically positioned to offer S-STEM scholarships to talented, financially disadvantaged students in the IT discipline. This article shares our experience and strategies from managing the ISU CS/IS Scholarship Program, a National Science Foundation (NSF) S-STEM scholarship grant. Leveraging our unique educational setting and multiple student support activities, we were able to provide financial support as well as implement several strategies needed to educate and retain qualified undergraduate IT students.
Let Zn denote the group of integers modulo n and let E (k) n be the set of all k-element subsets ... more Let Zn denote the group of integers modulo n and let E (k) n be the set of all k-element subsets of Zn where 1 ≤ k < n.
We answer the following question: If p and 4 are positive integers greater than 1 and C,, is the ... more We answer the following question: If p and 4 are positive integers greater than 1 and C,, is the set of all numbers in [0, 1 ] which can be expressed in base p without using a nonempty finite collection of finite length patterns in Z,, under what conditions does C, contain a number whose base q expansion contains all patterns of finite length? 0 1990 Academic press, Inc. Let p, k, and q be positive integers with p and q greater than 1. We define a p-pattern to be a finite block of digits in 2, (integers (mod p)). Let G= (PI, P,, . . . . Pk} be a collection of p-patterns. Let C, = {x E [O, 1 ] :x is expressible in base p without using any of the patterns in G}. We prove the following: (1) CP contains a Cantor set if and only if it contains an irrational, and (2) if C, contains a Cantor set, then C, contains a number whose base q expansion contains all q-patterns of finite length if and only if p and q are not powers of the same base. We also discuss some properties of C, and establish a relationship between the base q expansion of a number and the density of its orbit under a variation on the baker map with q branches. We will refer to the patterns in G as forbidden patterns. If M and N are finite length patterns, then M and N are similar if there exist positive integers m and n such that the pattern obtained by concatenating m blocks of the pattern M is the same as the pattern obtained by concatenating it blocks of the pattern N. A collection F of patterns is freely allowable if no concatenation of patterns in F contains a forbidden pattern. Throughout this paper, we will let E,(x) denote the nonterminating base p expansion of x. Also, E,(x) = (.x1 . . . xi-,xi . . . x,) means that the block xi . . . x, is repeated, ad infinitum, in the base p expansion of x. LEMMA 1. If there exist two nonsimilar freely allowable patterns, then C, contains a Cantor set.
It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of orde... more It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order v if and only if v ≡ 0, 1, or 2 (mod 9). For all positive integers λ and v, we find a maximum packing with loose 3-cycles of the λ-fold complete 3-uniform hypergraph of order v. We show that, if v ≥ 6, such a packing has a leave of two or fewer edges.
For integers r ≥ 2 and s ≥ 1, let K r×s denote the complete multipartite graph with r partite set... more For integers r ≥ 2 and s ≥ 1, let K r×s denote the complete multipartite graph with r partite sets of order s. Let G be a 2-regular graph of odd order n. If G contains exactly one odd cycle, it is known that there exists a G-decomposition of K 2kn+1 , of K (2k+1)×n , and of K k ×2n for all positive integers k and k ≥ 3. If G consists of three vertex-disjoint odd cycles, then the only known general result is a G-decomposition of K 2n+1. We use a novel extension of the Bose construction for triple systems to show that in the three odd cycles case, there exists a G-decomposition of K (2k+1)×n for every positive integer k. We also show that there exists a G-decomposition of K k×2n as well as of K 2kn+1 for every integer k ≥ 3.
We show how to obtain maximum packings of K,,, +v with k-cycles when k>3 is odd, g a positive int... more We show how to obtain maximum packings of K,,, +v with k-cycles when k>3 is odd, g a positive integer, and v even with O<v< 2k. Moreover, under certain conditions on v, we obtain maximum packings of Kzkg+".
We show that if G is a cubic graph of order 8, then there exists a G-decomposition of K-v if and ... more We show that if G is a cubic graph of order 8, then there exists a G-decomposition of K-v if and only if v 1 or 16 (mod 24).
There are ten bipartite cubic graphs of order n ≤ 12. For each such graph G we give necessary and... more There are ten bipartite cubic graphs of order n ≤ 12. For each such graph G we give necessary and sufficient conditions for the existence of decompositions of Kn and of Km,n into copies of G.
Bulletin of the Malaysian Mathematical Sciences Society, May 24, 2017
A common question in the study of graph decompositions is when does a graph G decompose the compl... more A common question in the study of graph decompositions is when does a graph G decompose the complete graph or the complete graph with a 1-factor removed or added. It is known that a $$\sigma $$σ-tripartite labeling of a tripartite graph G with n edges can be used to obtain a cyclic G-decomposition of $$K_{2nt+1}$$K2nt+1 for every positive integer t. Moreover, it can be used to obtain a cyclic G-decomposition of both $$K_{2nt+2}-I$$K2nt+2-I and $$K_{2nt}+I$$K2nt+I, where I is a 1-factor. We show that if G is an odd prism on 10 or more vertices or an even Möbius ladder, then G admits a $$\sigma $$σ-tripartite labeling.
Electronic Notes in Discrete Mathematics, Jul 1, 2017
It is known that for a given (simple) graph G with n edges, there exits a cyclic G-decomposition ... more It is known that for a given (simple) graph G with n edges, there exits a cyclic G-decomposition of K 2n+1 if and only if G admits a ρ-labeling. It is also known that if G is bipartite and it admits an ordered ρ-labeling, then there exists a cyclic G-decomposition of K 2nx+1 for every positive integer x. We extend these concepts to labelings of multigraphs through what we call λ-fold ρ-labelings and ordered λfold ρ-labelings. Let λ K m denote the λ-fold complete graph of order m. We show that if a subgraph G of λ K 2n/λ+1 has size n, there exits a cyclic G-decomposition of λ K 2n/λ+1 if and only if G admits a λ-fold ρ-labeling. If in addition G is bipartite and it admits an ordered λ-fold ρ-labeling, then there exists a cyclic G-decomposition of λ K 2nx/λ+1 for every positive integer x. We discuss some classes of graphs and multigraphs that admit such labelings.
Let D be any of the 10 digraphs obtained by orienting the edges of K 4 -e. We establish necessary... more Let D be any of the 10 digraphs obtained by orienting the edges of K 4 -e. We establish necessary and sufficient conditions for the existence of a (K * n , D)-design for 8 of these digraphs. Partial results as well as some nonexistence results are established for the remaining 2 digraphs.
For all integers m, nand t, we determine necessary and sufficient conditions for the existence of... more For all integers m, nand t, we determine necessary and sufficient conditions for the existence of (1) a pair of 3-cube decompositions of Kn having precisely t common 3-cubes; and (2) a pair of 3-cube decompositions of Km,n having precisely t common 3-cubes.
ABSTRACT We introduce the concept of a free α-valuation of a graph, and prove that the vertex-dis... more ABSTRACT We introduce the concept of a free α-valuation of a graph, and prove that the vertex-disjoint union of any collection of graphs with free α-valuations has an α-valuation. Many bipartite graphs have free α-valuations including the complete bipartite graph K m,n when m&gt;1 and n&gt;2, and the d-cube Q d for d&gt;2.
A k-factorization of the complete t-uniform hypergraph v . For v ≤ 9, we use nauty to generate th... more A k-factorization of the complete t-uniform hypergraph v . For v ≤ 9, we use nauty to generate the 2-regular and 3-regular spanning subhypergraphs of K (3) v and investigate which of these subhypergraphs factorize v -I, where I is a 1-factor. We settle this question for all but two of these subhypergraphs.
We show via an exhaustive computer search that there does not exist a (K 6 -e)decomposition of K ... more We show via an exhaustive computer search that there does not exist a (K 6 -e)decomposition of K 29 . This is the first example of a non-complete graph G for which a G-decomposition of K 2|E(G)|+1 does not exist.
For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on th... more For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on the number of cube and bipartition vertices must hold. We prove these necessary conditions sufficient in some cases. For cubes of fixed dimension d (indeed for d-regular bipartite graphs in general) we show that proving sufficiency can be reduced to decomposing a finite number of complete bipartite graphs. When t = 2d-' and r is the remainder on dividing t by d, we show Kr,r is decomposable into d-cubes and an r-factor, where if r > 0 this r-factor itself is decomposable into r-cubes.
Information Systems Education Journal, Jun 1, 2018
Our nation's competitive edge is highly dependent on the success of STEM education and the abilit... more Our nation's competitive edge is highly dependent on the success of STEM education and the ability of information technology (IT) graduates to find jobs. The School of Information Technology at Illinois State University (ISU) is strategically positioned to offer S-STEM scholarships to talented, financially disadvantaged students in the IT discipline. This article shares our experience and strategies from managing the ISU CS/IS Scholarship Program, a National Science Foundation (NSF) S-STEM scholarship grant. Leveraging our unique educational setting and multiple student support activities, we were able to provide financial support as well as implement several strategies needed to educate and retain qualified undergraduate IT students.
Let Zn denote the group of integers modulo n and let E (k) n be the set of all k-element subsets ... more Let Zn denote the group of integers modulo n and let E (k) n be the set of all k-element subsets of Zn where 1 ≤ k < n.
We answer the following question: If p and 4 are positive integers greater than 1 and C,, is the ... more We answer the following question: If p and 4 are positive integers greater than 1 and C,, is the set of all numbers in [0, 1 ] which can be expressed in base p without using a nonempty finite collection of finite length patterns in Z,, under what conditions does C, contain a number whose base q expansion contains all patterns of finite length? 0 1990 Academic press, Inc. Let p, k, and q be positive integers with p and q greater than 1. We define a p-pattern to be a finite block of digits in 2, (integers (mod p)). Let G= (PI, P,, . . . . Pk} be a collection of p-patterns. Let C, = {x E [O, 1 ] :x is expressible in base p without using any of the patterns in G}. We prove the following: (1) CP contains a Cantor set if and only if it contains an irrational, and (2) if C, contains a Cantor set, then C, contains a number whose base q expansion contains all q-patterns of finite length if and only if p and q are not powers of the same base. We also discuss some properties of C, and establish a relationship between the base q expansion of a number and the density of its orbit under a variation on the baker map with q branches. We will refer to the patterns in G as forbidden patterns. If M and N are finite length patterns, then M and N are similar if there exist positive integers m and n such that the pattern obtained by concatenating m blocks of the pattern M is the same as the pattern obtained by concatenating it blocks of the pattern N. A collection F of patterns is freely allowable if no concatenation of patterns in F contains a forbidden pattern. Throughout this paper, we will let E,(x) denote the nonterminating base p expansion of x. Also, E,(x) = (.x1 . . . xi-,xi . . . x,) means that the block xi . . . x, is repeated, ad infinitum, in the base p expansion of x. LEMMA 1. If there exist two nonsimilar freely allowable patterns, then C, contains a Cantor set.
It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of orde... more It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order v if and only if v ≡ 0, 1, or 2 (mod 9). For all positive integers λ and v, we find a maximum packing with loose 3-cycles of the λ-fold complete 3-uniform hypergraph of order v. We show that, if v ≥ 6, such a packing has a leave of two or fewer edges.
For integers r ≥ 2 and s ≥ 1, let K r×s denote the complete multipartite graph with r partite set... more For integers r ≥ 2 and s ≥ 1, let K r×s denote the complete multipartite graph with r partite sets of order s. Let G be a 2-regular graph of odd order n. If G contains exactly one odd cycle, it is known that there exists a G-decomposition of K 2kn+1 , of K (2k+1)×n , and of K k ×2n for all positive integers k and k ≥ 3. If G consists of three vertex-disjoint odd cycles, then the only known general result is a G-decomposition of K 2n+1. We use a novel extension of the Bose construction for triple systems to show that in the three odd cycles case, there exists a G-decomposition of K (2k+1)×n for every positive integer k. We also show that there exists a G-decomposition of K k×2n as well as of K 2kn+1 for every integer k ≥ 3.
We show how to obtain maximum packings of K,,, +v with k-cycles when k>3 is odd, g a positive int... more We show how to obtain maximum packings of K,,, +v with k-cycles when k>3 is odd, g a positive integer, and v even with O<v< 2k. Moreover, under certain conditions on v, we obtain maximum packings of Kzkg+".
We show that if G is a cubic graph of order 8, then there exists a G-decomposition of K-v if and ... more We show that if G is a cubic graph of order 8, then there exists a G-decomposition of K-v if and only if v 1 or 16 (mod 24).
There are ten bipartite cubic graphs of order n ≤ 12. For each such graph G we give necessary and... more There are ten bipartite cubic graphs of order n ≤ 12. For each such graph G we give necessary and sufficient conditions for the existence of decompositions of Kn and of Km,n into copies of G.
Bulletin of the Malaysian Mathematical Sciences Society, May 24, 2017
A common question in the study of graph decompositions is when does a graph G decompose the compl... more A common question in the study of graph decompositions is when does a graph G decompose the complete graph or the complete graph with a 1-factor removed or added. It is known that a $$\sigma $$σ-tripartite labeling of a tripartite graph G with n edges can be used to obtain a cyclic G-decomposition of $$K_{2nt+1}$$K2nt+1 for every positive integer t. Moreover, it can be used to obtain a cyclic G-decomposition of both $$K_{2nt+2}-I$$K2nt+2-I and $$K_{2nt}+I$$K2nt+I, where I is a 1-factor. We show that if G is an odd prism on 10 or more vertices or an even Möbius ladder, then G admits a $$\sigma $$σ-tripartite labeling.
Electronic Notes in Discrete Mathematics, Jul 1, 2017
It is known that for a given (simple) graph G with n edges, there exits a cyclic G-decomposition ... more It is known that for a given (simple) graph G with n edges, there exits a cyclic G-decomposition of K 2n+1 if and only if G admits a ρ-labeling. It is also known that if G is bipartite and it admits an ordered ρ-labeling, then there exists a cyclic G-decomposition of K 2nx+1 for every positive integer x. We extend these concepts to labelings of multigraphs through what we call λ-fold ρ-labelings and ordered λfold ρ-labelings. Let λ K m denote the λ-fold complete graph of order m. We show that if a subgraph G of λ K 2n/λ+1 has size n, there exits a cyclic G-decomposition of λ K 2n/λ+1 if and only if G admits a λ-fold ρ-labeling. If in addition G is bipartite and it admits an ordered λ-fold ρ-labeling, then there exists a cyclic G-decomposition of λ K 2nx/λ+1 for every positive integer x. We discuss some classes of graphs and multigraphs that admit such labelings.
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Papers by Saad El-Zanati