Geňa Hahn - Academia.edu (original) (raw)

Papers by Geňa Hahn

Research paper thumbnail of The Search-Time of a Graph

We consider the game of Cops and Robber played on finite and countably infinite connected graphs.... more We consider the game of Cops and Robber played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to the new parameter called the search-time of the graph. While the search-time is bounded above by the number of vertices, we prove an upper bound of half the number of vertices for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have search-time within a small additive constant of the number of vertices. We consider the ratio of the search-time to the number of vertices, and extend this notion of search-time density to infinite graphs. For the infinite random graph, the search-time density can be any real number in [0, 1]. We also consider the search-time when more than one cop is required to win. We show that for the fixed number of cops, the search-time can be calculated by polynomial algorithm, but it is NP-complete to decide, whether k cops can capture ...

Research paper thumbnail of On cop-win dynamic graphs

Discrete applied mathematics, Oct 1, 2024

Research paper thumbnail of Cycles and Rays

Springer eBooks, 1990

physlnl sCI'nces , vol. 301) 'PubllShld In coopiriflon wtth NATO ScI'ntHlc Affllrs DivISIon.' lSB... more physlnl sCI'nces , vol. 301) 'PubllShld In coopiriflon wtth NATO ScI'ntHlc Affllrs DivISIon.' lSBN•13,978-94-01!M71~S (alk, p,plr) I. Piths Ind cycl ..

Research paper thumbnail of The Automorphism Group of the Wreath Product of Directed Graphs

European Journal of Combinatorics, Sep 1, 1980

We give necessary and sufficient conditions for the automorphism group of the wreath product of d... more We give necessary and sufficient conditions for the automorphism group of the wreath product of directed graphs to be the wreath product of their respective automorphism groups. This gives a generalization of a theorem of Sabidussi. INTRODucnON In [2] we defined a wreath product of hypergraphs and found necessary and sufficient conditions for the automorphism group of such a product to be the wreath product of the automorphism groups of the component hypergraphs. This generalized the appropriate theorem for graphs due to Sabidussi, [3]. There is another natural extension of Sabidussi's theorem, namely to directed graphs. This is the subject of the present paper. DEFINITIONS For convenience-and to re-establish notation-we state even some very standard definitions. Let X be a set; by X(2) we denote the set of ordered pairs of distinct elements of X, by IXI the cardinality of X. If U and V are (non-empty) subsets which partition X we write X = U(; V. A directed graph (digraph) D = (V, E) consists of a set V of vertices (points) and a set E s;;; V(2) of (directed) edges. In this context we will usually write xy for (x, y) E V(2). To each directed edge xy of D we associate an undirected edge [xy] = {x, y}; thus [xy] = [yx]. With this we can talk about the underlying graph G(D) = (V, [E]) of D given by [E] = {[xy]lxy E E}; similarly we talk of the underlying edge [e] of a directed edge e. The complement of D is the digraph 15 = (V, E) with xy E E if and only if xye E. We say that D is connected if for every partition of V = X (; Y (and "partition" here always means a non-trivial one) there is an e E E such that [e] n X ;t' 4> ;t' [e) n Y. If X s;;; V we denote by D(X) = (X, E(X)) the subgraph of D induced (i.e., xy E E(X) iff xy E E whenever x, y E X) by X. Given two disjoint digraphs D1 = (VI. E 1) and D2 = (V 2 , E 2) we define their directed join (dijoin) D1 v D2 =

Research paper thumbnail of Lexicographic products with high reconstruction numbers

Discrete Mathematics, May 1, 2012

The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to... more The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to uniquely determine the graph up to isomorphism. Bollobás showed that almost all graphs have reconstruction number equal to three. McMullen and Radziszowski published a catalogue of all graphs on at most ten vertices with reconstruction number greater than three. We introduce constructions that generalize the examples identified in their work. In particular, we use lexicographic products of vertex transitive graphs with certain starter graphs from the work of Myrvold and from the work of Harary and Plantholt to generate new infinite families of graphs with high reconstruction numbers. In the process, we settle a question of McMullen and Radziszowski.

Research paper thumbnail of Cops and Robbers ordinals of cop-win trees

Discrete Mathematics, May 1, 2017

A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their s... more A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers. As a by-product of that characterization, each cop-win graph is assigned a unique ordinal, which we refer to as a CRordinal. For finite graphs, CR-ordinals correspond to the length of the game assuming optimal play, with the cop beginning the game in a least favourable initial position. For infinite graphs, however, the possible values of CR-ordinals have not been considered in the literature until the present work. We classify the CR-ordinals of cop-win trees as either a finite ordinal, or those of the form α + ω, where α is a limit ordinal. For general infinite cop-win graphs, we provide an example whose CR-ordinal is not of this form. We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs.

Research paper thumbnail of Cycle regular graphs need not be transitive

Discrete Applied Mathematics, Oct 1, 2001

We show that regular cycle-regular (see text for deÿnitions) graphs are not always vertex transit... more We show that regular cycle-regular (see text for deÿnitions) graphs are not always vertex transitive.

Research paper thumbnail of Infinite generalized friendship graphs

Discrete Mathematics, May 1, 1984

We give necessary and sufficient conditions for the existence of infinite generalized friendship ... more We give necessary and sufficient conditions for the existence of infinite generalized friendship graphs and show that there are 2 c non-isomorphic ones of each admissible order c and chromatic number. Further we prove that such graphs and their complements are almost always regular of degree equal to the order and that various generalizations of the Friendship Theorem do not hold for infinite generalized friendship graphs.

Research paper thumbnail of On a matrix partition conjecture

Journal of Combinatorial Theory, Series A, Feb 1, 1995

In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be part... more In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt× nt matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. We show that this conjecture is true for some values of m, n, t and k but that it is false in general.

Research paper thumbnail of Graph symmetry : algebraic methods and applications

Kluwer Academic Publishers eBooks, 1997

... of Computer Science & Mathematics Marist College Poughkeepsie, NY 12601-1387 ... more ... of Computer Science & Mathematics Marist College Poughkeepsie, NY 12601-1387 USA richard. goldstone@ marist. ... huji. ac. il Maria Flavia MAMMANA Dipartimento di Matematica Universita di Catania Viale A. Doria 6 1-95125 Catania Italy flavia@ dipmat. unict. ...

Research paper thumbnail of Short cycles in mediate graphs

Discrete Mathematics, 1983

We determine bounds for the smatlest f(n) such that every mediate graph with n vertices contains ... more We determine bounds for the smatlest f(n) such that every mediate graph with n vertices contains a (directed) cycle of length at most f(n).

Research paper thumbnail of Counting feasible solutions of the traveling salesman problem with pickups and deliveries is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>#</mml:mi><mml:mi>P</mml:mi></mml:math>-complete

Discrete Applied Mathematics, Jun 1, 2009

Research paper thumbnail of More star anti-Ramsey numbers

Research paper thumbnail of Some star anti-Ramsey numbers

Research paper thumbnail of The search-time of a graph

We consider the game of Cops and Robber played on finite and countably infinite connected graphs.... more We consider the game of Cops and Robber played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to the new param-eter called the search-time of the graph. While the search-time is bounded above by the number of vertices, we prove an upper bound of half the number of vertices for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have search-time within a small ad-ditive constant of the number of vertices. We consider the ratio of the search-time to the number of vertices, and extend this notion of search-time density to infinite graphs. For the infinite random graph, the search-time density can be any real number in [0, 1]. We also consider the search-time when more than one cop is re-quired to win. We show that for the fixed number of cops, the search-time can be calculated by polynomial algorithm, but it is NP-complete to decide, whether k cops can captu...

Research paper thumbnail of Special issue: Infinite graphs: Introductions, connections, surveys. Based on the workshop on infinite graphs, BIRS, Banff, Canada, October 2007

Research paper thumbnail of More star sub-ramsey numbers

Applied Mathematics and Computation, 1981

... 303–310. [4]; G. Hahn; Anti-ramsey numbers: An introduction. M.Sc. Thesis, Simon Fraser Unive... more ... 303–310. [4]; G. Hahn; Anti-ramsey numbers: An introduction. M.Sc. Thesis, Simon Fraser University, Burnaby, BC (1977). [5]; F. Harary; Graph Theory. Addison-Wesley, Don Mills (1969). Copyright © 1981 Published by Elsevier BV. Supplementary content. 1 2 3 4 5. I am the caption ...

Research paper thumbnail of A lower bound on the size of an absorbing set in an arc-coloured tournament

arXiv (Cornell University), Aug 29, 2017

Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer n, there is a l... more Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer n, there is a least integer f (n) such that any tournament whose arcs are coloured with n colours contains a subset of vertices S of size f (n) with the property that any vertex not in S admits a monochromatic path to some vertex of S. In this note we provide a lower bound on the value f (n).

Research paper thumbnail of Edge clique covers in graphs with independence number two

Journal of Graph Theory, Jan 21, 2021

The edge clique cover number ecc(G) of a graph G is size of the smallest collection of complete s... more The edge clique cover number ecc(G) of a graph G is size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw-free, then ecc(G) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for clawfree graphs with independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw-free. We give the first known proof of a linear bound in n for ecc(G) for such graphs, improving upon the bound of O (n 4/3 log 1/3 n) due to Javadi, Maleki and Omoomi. More precisely we prove that ecc(G) is at most the minimum of n + δ(G) and 2n − Ω(n log n), where δ(G) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example when the edge packing number with respect to cliques (a lower bound for ecc(G)) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.

Research paper thumbnail of Absorbing sets in arc-coloured tournaments*1

Discrete Mathematics, Jun 1, 2004

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of ... more Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deals with ÿnite tournaments, the last section treats inÿnite ones. In each case questions are suggested, both old and new.

Research paper thumbnail of The Search-Time of a Graph

We consider the game of Cops and Robber played on finite and countably infinite connected graphs.... more We consider the game of Cops and Robber played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to the new parameter called the search-time of the graph. While the search-time is bounded above by the number of vertices, we prove an upper bound of half the number of vertices for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have search-time within a small additive constant of the number of vertices. We consider the ratio of the search-time to the number of vertices, and extend this notion of search-time density to infinite graphs. For the infinite random graph, the search-time density can be any real number in [0, 1]. We also consider the search-time when more than one cop is required to win. We show that for the fixed number of cops, the search-time can be calculated by polynomial algorithm, but it is NP-complete to decide, whether k cops can capture ...

Research paper thumbnail of On cop-win dynamic graphs

Discrete applied mathematics, Oct 1, 2024

Research paper thumbnail of Cycles and Rays

Springer eBooks, 1990

physlnl sCI'nces , vol. 301) 'PubllShld In coopiriflon wtth NATO ScI'ntHlc Affllrs DivISIon.' lSB... more physlnl sCI'nces , vol. 301) 'PubllShld In coopiriflon wtth NATO ScI'ntHlc Affllrs DivISIon.' lSBN•13,978-94-01!M71~S (alk, p,plr) I. Piths Ind cycl ..

Research paper thumbnail of The Automorphism Group of the Wreath Product of Directed Graphs

European Journal of Combinatorics, Sep 1, 1980

We give necessary and sufficient conditions for the automorphism group of the wreath product of d... more We give necessary and sufficient conditions for the automorphism group of the wreath product of directed graphs to be the wreath product of their respective automorphism groups. This gives a generalization of a theorem of Sabidussi. INTRODucnON In [2] we defined a wreath product of hypergraphs and found necessary and sufficient conditions for the automorphism group of such a product to be the wreath product of the automorphism groups of the component hypergraphs. This generalized the appropriate theorem for graphs due to Sabidussi, [3]. There is another natural extension of Sabidussi's theorem, namely to directed graphs. This is the subject of the present paper. DEFINITIONS For convenience-and to re-establish notation-we state even some very standard definitions. Let X be a set; by X(2) we denote the set of ordered pairs of distinct elements of X, by IXI the cardinality of X. If U and V are (non-empty) subsets which partition X we write X = U(; V. A directed graph (digraph) D = (V, E) consists of a set V of vertices (points) and a set E s;;; V(2) of (directed) edges. In this context we will usually write xy for (x, y) E V(2). To each directed edge xy of D we associate an undirected edge [xy] = {x, y}; thus [xy] = [yx]. With this we can talk about the underlying graph G(D) = (V, [E]) of D given by [E] = {[xy]lxy E E}; similarly we talk of the underlying edge [e] of a directed edge e. The complement of D is the digraph 15 = (V, E) with xy E E if and only if xye E. We say that D is connected if for every partition of V = X (; Y (and "partition" here always means a non-trivial one) there is an e E E such that [e] n X ;t' 4> ;t' [e) n Y. If X s;;; V we denote by D(X) = (X, E(X)) the subgraph of D induced (i.e., xy E E(X) iff xy E E whenever x, y E X) by X. Given two disjoint digraphs D1 = (VI. E 1) and D2 = (V 2 , E 2) we define their directed join (dijoin) D1 v D2 =

Research paper thumbnail of Lexicographic products with high reconstruction numbers

Discrete Mathematics, May 1, 2012

The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to... more The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to uniquely determine the graph up to isomorphism. Bollobás showed that almost all graphs have reconstruction number equal to three. McMullen and Radziszowski published a catalogue of all graphs on at most ten vertices with reconstruction number greater than three. We introduce constructions that generalize the examples identified in their work. In particular, we use lexicographic products of vertex transitive graphs with certain starter graphs from the work of Myrvold and from the work of Harary and Plantholt to generate new infinite families of graphs with high reconstruction numbers. In the process, we settle a question of McMullen and Radziszowski.

Research paper thumbnail of Cops and Robbers ordinals of cop-win trees

Discrete Mathematics, May 1, 2017

A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their s... more A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers. As a by-product of that characterization, each cop-win graph is assigned a unique ordinal, which we refer to as a CRordinal. For finite graphs, CR-ordinals correspond to the length of the game assuming optimal play, with the cop beginning the game in a least favourable initial position. For infinite graphs, however, the possible values of CR-ordinals have not been considered in the literature until the present work. We classify the CR-ordinals of cop-win trees as either a finite ordinal, or those of the form α + ω, where α is a limit ordinal. For general infinite cop-win graphs, we provide an example whose CR-ordinal is not of this form. We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs.

Research paper thumbnail of Cycle regular graphs need not be transitive

Discrete Applied Mathematics, Oct 1, 2001

We show that regular cycle-regular (see text for deÿnitions) graphs are not always vertex transit... more We show that regular cycle-regular (see text for deÿnitions) graphs are not always vertex transitive.

Research paper thumbnail of Infinite generalized friendship graphs

Discrete Mathematics, May 1, 1984

We give necessary and sufficient conditions for the existence of infinite generalized friendship ... more We give necessary and sufficient conditions for the existence of infinite generalized friendship graphs and show that there are 2 c non-isomorphic ones of each admissible order c and chromatic number. Further we prove that such graphs and their complements are almost always regular of degree equal to the order and that various generalizations of the Friendship Theorem do not hold for infinite generalized friendship graphs.

Research paper thumbnail of On a matrix partition conjecture

Journal of Combinatorial Theory, Series A, Feb 1, 1995

In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be part... more In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt× nt matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. We show that this conjecture is true for some values of m, n, t and k but that it is false in general.

Research paper thumbnail of Graph symmetry : algebraic methods and applications

Kluwer Academic Publishers eBooks, 1997

... of Computer Science &amp;amp;amp; Mathematics Marist College Poughkeepsie, NY 12601-1387 ... more ... of Computer Science &amp;amp;amp; Mathematics Marist College Poughkeepsie, NY 12601-1387 USA richard. goldstone@ marist. ... huji. ac. il Maria Flavia MAMMANA Dipartimento di Matematica Universita di Catania Viale A. Doria 6 1-95125 Catania Italy flavia@ dipmat. unict. ...

Research paper thumbnail of Short cycles in mediate graphs

Discrete Mathematics, 1983

We determine bounds for the smatlest f(n) such that every mediate graph with n vertices contains ... more We determine bounds for the smatlest f(n) such that every mediate graph with n vertices contains a (directed) cycle of length at most f(n).

Research paper thumbnail of Counting feasible solutions of the traveling salesman problem with pickups and deliveries is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>#</mml:mi><mml:mi>P</mml:mi></mml:math>-complete

Discrete Applied Mathematics, Jun 1, 2009

Research paper thumbnail of More star anti-Ramsey numbers

Research paper thumbnail of Some star anti-Ramsey numbers

Research paper thumbnail of The search-time of a graph

We consider the game of Cops and Robber played on finite and countably infinite connected graphs.... more We consider the game of Cops and Robber played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to the new param-eter called the search-time of the graph. While the search-time is bounded above by the number of vertices, we prove an upper bound of half the number of vertices for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have search-time within a small ad-ditive constant of the number of vertices. We consider the ratio of the search-time to the number of vertices, and extend this notion of search-time density to infinite graphs. For the infinite random graph, the search-time density can be any real number in [0, 1]. We also consider the search-time when more than one cop is re-quired to win. We show that for the fixed number of cops, the search-time can be calculated by polynomial algorithm, but it is NP-complete to decide, whether k cops can captu...

Research paper thumbnail of Special issue: Infinite graphs: Introductions, connections, surveys. Based on the workshop on infinite graphs, BIRS, Banff, Canada, October 2007

Research paper thumbnail of More star sub-ramsey numbers

Applied Mathematics and Computation, 1981

... 303–310. [4]; G. Hahn; Anti-ramsey numbers: An introduction. M.Sc. Thesis, Simon Fraser Unive... more ... 303–310. [4]; G. Hahn; Anti-ramsey numbers: An introduction. M.Sc. Thesis, Simon Fraser University, Burnaby, BC (1977). [5]; F. Harary; Graph Theory. Addison-Wesley, Don Mills (1969). Copyright © 1981 Published by Elsevier BV. Supplementary content. 1 2 3 4 5. I am the caption ...

Research paper thumbnail of A lower bound on the size of an absorbing set in an arc-coloured tournament

arXiv (Cornell University), Aug 29, 2017

Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer n, there is a l... more Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer n, there is a least integer f (n) such that any tournament whose arcs are coloured with n colours contains a subset of vertices S of size f (n) with the property that any vertex not in S admits a monochromatic path to some vertex of S. In this note we provide a lower bound on the value f (n).

Research paper thumbnail of Edge clique covers in graphs with independence number two

Journal of Graph Theory, Jan 21, 2021

The edge clique cover number ecc(G) of a graph G is size of the smallest collection of complete s... more The edge clique cover number ecc(G) of a graph G is size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw-free, then ecc(G) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for clawfree graphs with independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw-free. We give the first known proof of a linear bound in n for ecc(G) for such graphs, improving upon the bound of O (n 4/3 log 1/3 n) due to Javadi, Maleki and Omoomi. More precisely we prove that ecc(G) is at most the minimum of n + δ(G) and 2n − Ω(n log n), where δ(G) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example when the edge packing number with respect to cliques (a lower bound for ecc(G)) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.

Research paper thumbnail of Absorbing sets in arc-coloured tournaments*1

Discrete Mathematics, Jun 1, 2004

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of ... more Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deals with ÿnite tournaments, the last section treats inÿnite ones. In each case questions are suggested, both old and new.