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Papers by Peter Hamburger

Research paper thumbnail of Hypercube sub-graphs with minimal detours

Journal of Graph Theory, 1996

Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter ... more Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter Hamburger * Raymond E. Pippert William D. Weakley Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, Indiana 46805 ...

Research paper thumbnail of A Symmetrical Beauty. A Non-simple 7-Venn Diagram With A Minimum Vertex Set

Ars Combinatoria, 2003

In this short note using the method developed in [10]and [11], we construct a highly symmetrical,... more In this short note using the method developed in [10]and [11], we construct a highly symmetrical, non-simple, attractive?-Venn diagram. This diagram has the minimum number of vertices,21. The only similar two, published in [1] and [11], differ fi'om ours inmany ways. One of them was found by computer search ([1]). Bothof them are "necklace" type Venn diagrams (see [14] for

Research paper thumbnail of A note on the asymmetry of rotating, stratified flow past a shallow ridge

Physics of Fluids A: Fluid Dynamics, 1993

Streamline patterns in experiments of Boyer and Biolley [Philos. Trans. R. Soc. London Ser. A 318... more Streamline patterns in experiments of Boyer and Biolley [Philos. Trans. R. Soc. London Ser. A 318, 411 (1986)] on flow past a shallow ridge on the floor of a channel in a rotating, stratified fluid exhibit a clear upstream/downstream asymmetry. In this note this question is considered and also a condition for the existence of an eddy in such flows is derived by examining the mathematical properties of the solution.

Research paper thumbnail of Analysis of Venn Diagrams Using Cycles in Graphs

This paper is the last in a series by the authors on the use of graph theory to analyze Venn diag... more This paper is the last in a series by the authors on the use of graph theory to analyze Venn diagrams on few curves. We complete the construction (and hence the enumeration) of spherical Venn diagrams on five curves, which yields additional results about conjectures of Grünbaum concerning which Venn diagrams are convex, which are exposed, and which can be

Research paper thumbnail of Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Util... more Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple,

Research paper thumbnail of On a leverage problem in the hypercube

Research paper thumbnail of Hypercube subgraphs with minimal detours

Journal of Graph Theory, 1996

Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter ... more Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter Hamburger * Raymond E. Pippert William D. Weakley Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, Indiana 46805 ...

Research paper thumbnail of Solution of fink & straight conjecture on path-perfect complete bipartite graphs

Journal of Graph Theory, 2007

We consider various edge disjoint partitions of complete bipartite graphs. One case is where we d... more We consider various edge disjoint partitions of complete bipartite graphs. One case is where we decompose the edge set into edge disjoint paths of increasing lengths. A graph G is path-perfect if there is a positive integer n such that the edge set E(G) of the graph G can be partitioned into paths of length 1; 2; 3; : : : ; n: The main re- sult of the paper is the proof of the conjecture of Fink and Straight (4): A com- plete bipartite graph Ks;t on t + s vertices (t s) is path-perfect if and only if there is a positive integer n such that the following two conditions are satised; (i) st = 1 + 2 + + n = n+1 2 , and (ii) n 2t: Our proof gives a linear time algorithm to nd an edge disjoint partition of a complete bipartite graph into paths of lengths 1; 2; : : : ; n.

Research paper thumbnail of Hypercube subgraphs with local detours

Journal of Graph Theory, 1999

Research paper thumbnail of Hamilton Cycles in Planar Graphs and Venn Diagrams

Journal of Combinatorial Theory, Series B, 1996

Abstract: Using graph theory, we prove Grünbaum's conjecture [4]: Every,Venn diagram,of n... more Abstract: Using graph theory, we prove Grünbaum's conjecture [4]: Every,Venn diagram,of n curves,can,be extended,to a Venn diagram,of n+1 curves,by the,addition,of a suitable simple closed curve. 1., Introduction:A,Venn diagram,consists,of n simple,closed,curves inthe,plane,so that all possible intersections (2,many),of the interiorsand the exteriors of these curves are nonempty and are connected.,One can put various restrictions on the diagrams and obtain specialclasses,of Venn,diagrams,[4]. The

Research paper thumbnail of Venn diagrams and planar graphs

Geometriae Dedicata, 1996

Venn diagrams are named after logician John Venn (see [4] for more information). In words, a Venn... more Venn diagrams are named after logician John Venn (see [4] for more information). In words, a Venn diagram consists of n simple closed curves in the plane so that all possible (2 n) intersections of the interiors and the exteriors of these curves are nonempty and are connected. One ...

Research paper thumbnail of Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs

European Journal of Combinatorics, 1999

ABSTRACT

Research paper thumbnail of Random packings of graphs

Discrete Mathematics, 1994

A graph G is said to be packable by the graph F if its edges can be partitioned into copies of F.... more A graph G is said to be packable by the graph F if its edges can be partitioned into copies of F. It is called randomly packable if what remains after deletion of the edges of a proper subgraph that is F-packable is also F-packable.

Research paper thumbnail of Random packing by ρ-connected ρ-regular graphs

Discrete Mathematics, 1996

ABSTRACT

Research paper thumbnail of The integrity of the cube is small

It is shown that the integrity of the n-dimensional cube is O(2n logn/ p n). Barefoot, Entringer ... more It is shown that the integrity of the n-dimensional cube is O(2n logn/ p n). Barefoot, Entringer and Swart (1) introduced the graphical parameter integrity. Let m(H) denote the maximum number of vertices in a component of a graph H. Then the integrity of a graph G, denoted I(G), is defined by I(G) = min{|S| + m(G S) : SV (G)}. In (2) the values of the integrity of some product graphs were calculated. One such graph for which the integrity was not calculated was the cube. The n- dimensional cube Qn has 2n vertices, and may be defined as the cartesian product of n copies of Q1, where Q1 is the complete graph on 2 vertices. In (2) it was conjectured that I(Qn) was 2n 1+1. We show here that this value is considerably too high.

Research paper thumbnail of Hypercube sub-graphs with minimal detours

Journal of Graph Theory, 1996

Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter ... more Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter Hamburger * Raymond E. Pippert William D. Weakley Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, Indiana 46805 ...

Research paper thumbnail of A Symmetrical Beauty. A Non-simple 7-Venn Diagram With A Minimum Vertex Set

Ars Combinatoria, 2003

In this short note using the method developed in [10]and [11], we construct a highly symmetrical,... more In this short note using the method developed in [10]and [11], we construct a highly symmetrical, non-simple, attractive?-Venn diagram. This diagram has the minimum number of vertices,21. The only similar two, published in [1] and [11], differ fi'om ours inmany ways. One of them was found by computer search ([1]). Bothof them are "necklace" type Venn diagrams (see [14] for

Research paper thumbnail of A note on the asymmetry of rotating, stratified flow past a shallow ridge

Physics of Fluids A: Fluid Dynamics, 1993

Streamline patterns in experiments of Boyer and Biolley [Philos. Trans. R. Soc. London Ser. A 318... more Streamline patterns in experiments of Boyer and Biolley [Philos. Trans. R. Soc. London Ser. A 318, 411 (1986)] on flow past a shallow ridge on the floor of a channel in a rotating, stratified fluid exhibit a clear upstream/downstream asymmetry. In this note this question is considered and also a condition for the existence of an eddy in such flows is derived by examining the mathematical properties of the solution.

Research paper thumbnail of Analysis of Venn Diagrams Using Cycles in Graphs

This paper is the last in a series by the authors on the use of graph theory to analyze Venn diag... more This paper is the last in a series by the authors on the use of graph theory to analyze Venn diagrams on few curves. We complete the construction (and hence the enumeration) of spherical Venn diagrams on five curves, which yields additional results about conjectures of Grünbaum concerning which Venn diagrams are convex, which are exposed, and which can be

Research paper thumbnail of Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Util... more Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple,

Research paper thumbnail of On a leverage problem in the hypercube

Research paper thumbnail of Hypercube subgraphs with minimal detours

Journal of Graph Theory, 1996

Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter ... more Page 1. Hypercube Subgraphs With Minimal Detours Péal Erdios Hungarian Academy of Sciences Peter Hamburger * Raymond E. Pippert William D. Weakley Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, Indiana 46805 ...

Research paper thumbnail of Solution of fink & straight conjecture on path-perfect complete bipartite graphs

Journal of Graph Theory, 2007

We consider various edge disjoint partitions of complete bipartite graphs. One case is where we d... more We consider various edge disjoint partitions of complete bipartite graphs. One case is where we decompose the edge set into edge disjoint paths of increasing lengths. A graph G is path-perfect if there is a positive integer n such that the edge set E(G) of the graph G can be partitioned into paths of length 1; 2; 3; : : : ; n: The main re- sult of the paper is the proof of the conjecture of Fink and Straight (4): A com- plete bipartite graph Ks;t on t + s vertices (t s) is path-perfect if and only if there is a positive integer n such that the following two conditions are satised; (i) st = 1 + 2 + + n = n+1 2 , and (ii) n 2t: Our proof gives a linear time algorithm to nd an edge disjoint partition of a complete bipartite graph into paths of lengths 1; 2; : : : ; n.

Research paper thumbnail of Hypercube subgraphs with local detours

Journal of Graph Theory, 1999

Research paper thumbnail of Hamilton Cycles in Planar Graphs and Venn Diagrams

Journal of Combinatorial Theory, Series B, 1996

Abstract: Using graph theory, we prove Grünbaum's conjecture [4]: Every,Venn diagram,of n... more Abstract: Using graph theory, we prove Grünbaum's conjecture [4]: Every,Venn diagram,of n curves,can,be extended,to a Venn diagram,of n+1 curves,by the,addition,of a suitable simple closed curve. 1., Introduction:A,Venn diagram,consists,of n simple,closed,curves inthe,plane,so that all possible intersections (2,many),of the interiorsand the exteriors of these curves are nonempty and are connected.,One can put various restrictions on the diagrams and obtain specialclasses,of Venn,diagrams,[4]. The

Research paper thumbnail of Venn diagrams and planar graphs

Geometriae Dedicata, 1996

Venn diagrams are named after logician John Venn (see [4] for more information). In words, a Venn... more Venn diagrams are named after logician John Venn (see [4] for more information). In words, a Venn diagram consists of n simple closed curves in the plane so that all possible (2 n) intersections of the interiors and the exteriors of these curves are nonempty and are connected. One ...

Research paper thumbnail of Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs

European Journal of Combinatorics, 1999

ABSTRACT

Research paper thumbnail of Random packings of graphs

Discrete Mathematics, 1994

A graph G is said to be packable by the graph F if its edges can be partitioned into copies of F.... more A graph G is said to be packable by the graph F if its edges can be partitioned into copies of F. It is called randomly packable if what remains after deletion of the edges of a proper subgraph that is F-packable is also F-packable.

Research paper thumbnail of Random packing by ρ-connected ρ-regular graphs

Discrete Mathematics, 1996

ABSTRACT

Research paper thumbnail of The integrity of the cube is small

It is shown that the integrity of the n-dimensional cube is O(2n logn/ p n). Barefoot, Entringer ... more It is shown that the integrity of the n-dimensional cube is O(2n logn/ p n). Barefoot, Entringer and Swart (1) introduced the graphical parameter integrity. Let m(H) denote the maximum number of vertices in a component of a graph H. Then the integrity of a graph G, denoted I(G), is defined by I(G) = min{|S| + m(G S) : SV (G)}. In (2) the values of the integrity of some product graphs were calculated. One such graph for which the integrity was not calculated was the cube. The n- dimensional cube Qn has 2n vertices, and may be defined as the cartesian product of n copies of Q1, where Q1 is the complete graph on 2 vertices. In (2) it was conjectured that I(Qn) was 2n 1+1. We show here that this value is considerably too high.