Paul Kainen | Georgetown University (original) (raw)
Papers by Paul Kainen
Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any f... more Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. We prove that for each positive integer k there exist finite k-superuniversal graphs, and e find upper and lo er bounds on the smallest such graphs. We also find various bounds on the number of edges as ell as the maximal and minimal valence of a k-superuniversal graph. We then generalize the notion of k-superuniversality to cover graphs ith colorings and prove similar and related theorems. 1
We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs ... more We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.
SSRN Electronic Journal, 2022
The matching book thickness mbt(G) of a graph G is the least number of pages in a book embedding ... more The matching book thickness mbt(G) of a graph G is the least number of pages in a book embedding such that each page is a matching; G is dispersable if mbt(G) = ∆(G), where ∆ is the maximum degree. A graph G is nearly dispersable if mbt(G) = 1 + ∆(G). Recently, Alam et al disproved the 40-year-old Bernhart-Kainen conjecture that all regular bipartite graphs are dispersable, motivating further work on dispersability. We show that most circulant graphs G with jump lengths not exceeding 3 are dispersable or nearly dispersable. Similar results are obtained for arbitrarily large jump lengths, obtained by rearranging repeated copies of a complete or bipartite complete graph.
It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×... more It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×Qs has a quadrilateral embedding in some surface, where Qs is the hypercube graph. This answers a question of Pisanski.
It is shown that the number of pages required for a book embedding of a graph is the maximum of t... more It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for X-trees and square grids.
Just as a busy kitchen can be more efficient than an idle one, Kleinrock showed 35 years ago that... more Just as a busy kitchen can be more efficient than an idle one, Kleinrock showed 35 years ago that heavily used networks admit simple heuristic approximations with excellent quantitative accuracy. We describe a number of different examples in which having many parameters actually facilitates computation and we suggest connections with geometric phenomena in high-dimensional spaces. It seems that in several interesting and quite general situations, dimensionality may be a blessing in disguise provided that some suitable form of computing is used which can deal with it. 1. Introduction This is a particularly appropriate time to consider the problem of systems having a very large number of parameters. Reasons include: neural networks, genetic algorithms, expert systems, fuzzy logic and cellular automata. Moreover, the current level of sensor technology can flood us with data like water from a firehose. Already, there are terrabytes of data that have been collected by NASA and others whi...
The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a m... more The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread.
In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions co... more In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. Keywords. One-hidden-layer networks, Heaviside perceptrons, best approximation, metric projection, continuous selection, approximatively compact. 1
We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs ... more We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.
The American Mathematical Monthly, 2021
Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-... more Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.
A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 +... more A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l < k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.
Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both... more Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...
Theoretical results on approximation of multivariable functions by feedforward neural networks ar... more Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.
On montre que, dans toute collection non vide d'au plus d-2 carres d'un hypercube Q d de ... more On montre que, dans toute collection non vide d'au plus d-2 carres d'un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d'isomorphismes sur le schema de l'hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.
Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both... more Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...
Issues for transport facilities on the lunar surface related to science, engineering, architectur... more Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. Logistic decisions made in the next decade may be crucial to financial success. In addition to outlining some of the problems and their relations with math and computation, the paper provides useful resources for decision-makers, scientists, and engineers. Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.
A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 ... more A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.
The Art of Discrete and Applied Mathematics
It is shown that the 2-skeleton of the odd-d-dimensional hypercube can be decomposed into s d sph... more It is shown that the 2-skeleton of the odd-d-dimensional hypercube can be decomposed into s d spheres and τ d tori, where s d = (d − 1)2 d−4 and τ d is asymptotically in the range (64/9)2 d−7 to (d − 1)(d − 3)2 d−7 .
The Art of Discrete and Applied Mathematics
For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the unio... more For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the union of pairwise face-disjoint isomorphic 2-complexes, each a topological sphere. If d = 5 n , then such a decomposition can be achieved, but with non-isomorphic spheres.
Studies in Computational Intelligence
Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any f... more Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. We prove that for each positive integer k there exist finite k-superuniversal graphs, and e find upper and lo er bounds on the smallest such graphs. We also find various bounds on the number of edges as ell as the maximal and minimal valence of a k-superuniversal graph. We then generalize the notion of k-superuniversality to cover graphs ith colorings and prove similar and related theorems. 1
We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs ... more We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.
SSRN Electronic Journal, 2022
The matching book thickness mbt(G) of a graph G is the least number of pages in a book embedding ... more The matching book thickness mbt(G) of a graph G is the least number of pages in a book embedding such that each page is a matching; G is dispersable if mbt(G) = ∆(G), where ∆ is the maximum degree. A graph G is nearly dispersable if mbt(G) = 1 + ∆(G). Recently, Alam et al disproved the 40-year-old Bernhart-Kainen conjecture that all regular bipartite graphs are dispersable, motivating further work on dispersability. We show that most circulant graphs G with jump lengths not exceeding 3 are dispersable or nearly dispersable. Similar results are obtained for arbitrarily large jump lengths, obtained by rearranging repeated copies of a complete or bipartite complete graph.
It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×... more It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×Qs has a quadrilateral embedding in some surface, where Qs is the hypercube graph. This answers a question of Pisanski.
It is shown that the number of pages required for a book embedding of a graph is the maximum of t... more It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for X-trees and square grids.
Just as a busy kitchen can be more efficient than an idle one, Kleinrock showed 35 years ago that... more Just as a busy kitchen can be more efficient than an idle one, Kleinrock showed 35 years ago that heavily used networks admit simple heuristic approximations with excellent quantitative accuracy. We describe a number of different examples in which having many parameters actually facilitates computation and we suggest connections with geometric phenomena in high-dimensional spaces. It seems that in several interesting and quite general situations, dimensionality may be a blessing in disguise provided that some suitable form of computing is used which can deal with it. 1. Introduction This is a particularly appropriate time to consider the problem of systems having a very large number of parameters. Reasons include: neural networks, genetic algorithms, expert systems, fuzzy logic and cellular automata. Moreover, the current level of sensor technology can flood us with data like water from a firehose. Already, there are terrabytes of data that have been collected by NASA and others whi...
The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a m... more The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread.
In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions co... more In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. Keywords. One-hidden-layer networks, Heaviside perceptrons, best approximation, metric projection, continuous selection, approximatively compact. 1
We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs ... more We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.
The American Mathematical Monthly, 2021
Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-... more Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.
A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 +... more A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l < k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.
Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both... more Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...
Theoretical results on approximation of multivariable functions by feedforward neural networks ar... more Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.
On montre que, dans toute collection non vide d'au plus d-2 carres d'un hypercube Q d de ... more On montre que, dans toute collection non vide d'au plus d-2 carres d'un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d'isomorphismes sur le schema de l'hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.
Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both... more Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...
Issues for transport facilities on the lunar surface related to science, engineering, architectur... more Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. Logistic decisions made in the next decade may be crucial to financial success. In addition to outlining some of the problems and their relations with math and computation, the paper provides useful resources for decision-makers, scientists, and engineers. Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.
A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 ... more A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.
The Art of Discrete and Applied Mathematics
It is shown that the 2-skeleton of the odd-d-dimensional hypercube can be decomposed into s d sph... more It is shown that the 2-skeleton of the odd-d-dimensional hypercube can be decomposed into s d spheres and τ d tori, where s d = (d − 1)2 d−4 and τ d is asymptotically in the range (64/9)2 d−7 to (d − 1)(d − 3)2 d−7 .
The Art of Discrete and Applied Mathematics
For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the unio... more For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the union of pairwise face-disjoint isomorphic 2-complexes, each a topological sphere. If d = 5 n , then such a decomposition can be achieved, but with non-isomorphic spheres.
Studies in Computational Intelligence