Glenn Hopkins - Academia.edu (original) (raw)

Papers by Glenn Hopkins

Research paper thumbnail of An algebraic approach to the residue symbol

Research paper thumbnail of Maximal bipartite subgraphs in triangle-free graphs

… of the thirteenth …, 1982

Research paper thumbnail of An algebraic approach to Grothendieck’s residue symbol

Transactions of the American Mathematical Society, 1983

A certain map—the "residue map"—is defined and its properties are investigated. The imp... more A certain map—the "residue map"—is defined and its properties are investigated. The impetus for the definition and study of this map is a definition by A. Grothendieck of a homomorphism, the "residue symbol", which has been found applicable in several areas, including the duality theory of algebraic varieties.

Research paper thumbnail of Extremal bipartite subgraphs of cubic triangle-free graphs

Journal of Graph Theory, 1982

A cubic triangle-free graph has a bipartite subgraph with at least 4/5 of the original edges. Exa... more A cubic triangle-free graph has a bipartite subgraph with at least 4/5 of the original edges. Examples show that this is a best possible result. ... It is easy to verify that the Petersen graph may be reduced to a bipartite graph by the removal of three of its 15 edges, and that this may not be done ...

Research paper thumbnail of On the number of cut-vertices in a graph

International Journal of Mathematics and Mathematical Sciences, 1989

A connected graph with n vertices contains no more thanr2r-2(n-2)cutvertices of degree r. All gra... more A connected graph with n vertices contains no more thanr2r-2(n-2)cutvertices of degree r. All graphs in which the bound is achieved are described. In addition, for graphs of maximum degree three and minimumδ, best possible bounds are obtained forδ=1, 2, 3.

Research paper thumbnail of Graphs with unique maximum independent sets

Discrete Mathematics, 1985

A graph is a unique independence graph if it has a unique maximum independent set. If, further, t... more A graph is a unique independence graph if it has a unique maximum independent set. If, further, the complement of the maximum independent set is independent, the graph is a strong unique independence graph. We characterize strong unique independence graphs and unique independence trees.

Research paper thumbnail of Girth and independence ratio

Canadian Mathematical Bulletin, 1982

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum ... more Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.

Research paper thumbnail of Lower bounds for induced forests in cubic graphs

Canadian Mathematical Bulletin, 1987

If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest con... more If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

Research paper thumbnail of On the independence dimension of a graph

Congressus numerantium, 1980

Research paper thumbnail of Commutative semigroup rings which are principal ideal rings

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE

Research paper thumbnail of Some Identities Arising from the Fibonacci Numbers of Certain Graphs

Research paper thumbnail of Outerplanarity without topology

Chartrand and Harary showed that, with the exception of K 4 with one edge removed, a graph is out... more Chartrand and Harary showed that, with the exception of K 4 with one edge removed, a graph is outerplanar if and only if it has no subgraph homeomorphic to either K 4 or K 2,3 . In the present note, the authors show that a non-trivial graph is maximal outerplanar (outerplanar, but the addition of any new edge destroys outerplanarity) if and only if it is 2-degenerate (every subgraph has minimum degree at most 2) and the neighbors of every vertex induce a path.

Research paper thumbnail of An identity arising from counting independent sets

Research paper thumbnail of Maximal bipartite subgraphs

Research paper thumbnail of Some identities arising from the Fibonacci numbers of certain graphs

Research paper thumbnail of Vertex partitions and k-small subsets of graphs

Research paper thumbnail of On the number of cut-vertices in a graph

International Journal of Mathematics and Mathematical Sciences, 1989

A connected graph with n vertices contains no more than -z(n-2) cutvertices of degree r. All grap... more A connected graph with n vertices contains no more than -z(n-2) cutvertices of degree r. All graphs in which the bound is achieved are described. In addition, for graphs of maximum degree three and minimum 6, best possible bounds are obtained for I, 2, 3.

Research paper thumbnail of An algebraic approach to the residue symbol

Research paper thumbnail of Maximal bipartite subgraphs in triangle-free graphs

… of the thirteenth …, 1982

Research paper thumbnail of An algebraic approach to Grothendieck’s residue symbol

Transactions of the American Mathematical Society, 1983

A certain map—the "residue map"—is defined and its properties are investigated. The imp... more A certain map—the "residue map"—is defined and its properties are investigated. The impetus for the definition and study of this map is a definition by A. Grothendieck of a homomorphism, the "residue symbol", which has been found applicable in several areas, including the duality theory of algebraic varieties.

Research paper thumbnail of Extremal bipartite subgraphs of cubic triangle-free graphs

Journal of Graph Theory, 1982

A cubic triangle-free graph has a bipartite subgraph with at least 4/5 of the original edges. Exa... more A cubic triangle-free graph has a bipartite subgraph with at least 4/5 of the original edges. Examples show that this is a best possible result. ... It is easy to verify that the Petersen graph may be reduced to a bipartite graph by the removal of three of its 15 edges, and that this may not be done ...

Research paper thumbnail of On the number of cut-vertices in a graph

International Journal of Mathematics and Mathematical Sciences, 1989

A connected graph with n vertices contains no more thanr2r-2(n-2)cutvertices of degree r. All gra... more A connected graph with n vertices contains no more thanr2r-2(n-2)cutvertices of degree r. All graphs in which the bound is achieved are described. In addition, for graphs of maximum degree three and minimumδ, best possible bounds are obtained forδ=1, 2, 3.

Research paper thumbnail of Graphs with unique maximum independent sets

Discrete Mathematics, 1985

A graph is a unique independence graph if it has a unique maximum independent set. If, further, t... more A graph is a unique independence graph if it has a unique maximum independent set. If, further, the complement of the maximum independent set is independent, the graph is a strong unique independence graph. We characterize strong unique independence graphs and unique independence trees.

Research paper thumbnail of Girth and independence ratio

Canadian Mathematical Bulletin, 1982

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum ... more Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.

Research paper thumbnail of Lower bounds for induced forests in cubic graphs

Canadian Mathematical Bulletin, 1987

If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest con... more If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

Research paper thumbnail of On the independence dimension of a graph

Congressus numerantium, 1980

Research paper thumbnail of Commutative semigroup rings which are principal ideal rings

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE

Research paper thumbnail of Some Identities Arising from the Fibonacci Numbers of Certain Graphs

Research paper thumbnail of Outerplanarity without topology

Chartrand and Harary showed that, with the exception of K 4 with one edge removed, a graph is out... more Chartrand and Harary showed that, with the exception of K 4 with one edge removed, a graph is outerplanar if and only if it has no subgraph homeomorphic to either K 4 or K 2,3 . In the present note, the authors show that a non-trivial graph is maximal outerplanar (outerplanar, but the addition of any new edge destroys outerplanarity) if and only if it is 2-degenerate (every subgraph has minimum degree at most 2) and the neighbors of every vertex induce a path.

Research paper thumbnail of An identity arising from counting independent sets

Research paper thumbnail of Maximal bipartite subgraphs

Research paper thumbnail of Some identities arising from the Fibonacci numbers of certain graphs

Research paper thumbnail of Vertex partitions and k-small subsets of graphs

Research paper thumbnail of On the number of cut-vertices in a graph

International Journal of Mathematics and Mathematical Sciences, 1989

A connected graph with n vertices contains no more than -z(n-2) cutvertices of degree r. All grap... more A connected graph with n vertices contains no more than -z(n-2) cutvertices of degree r. All graphs in which the bound is achieved are described. In addition, for graphs of maximum degree three and minimum 6, best possible bounds are obtained for I, 2, 3.