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Articles 31 - 42 of 42

Full-Text Articles in Discrete Mathematics and Combinatorics

Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West Apr 2012

Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West

Noah Prince

A Roman dominating function of a graph G is a labeling f: V(G) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ʋϵV(G)f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(G) + …


Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince Nov 2011

Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince

Hua Kun Liu

Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …


The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz Dec 2008

The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz

Noah Prince

Erdös and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.


Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince Dec 2007

Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince

Noah Prince

Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …


On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince Dec 2007

On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince

Noah Prince

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and …


On The Gauge Equivalence Of Twisted Quantum Doubles Of Elementary Abelian And Extra-Special 2-Groups, Christopher Goff, Geoffrey Mason, Siu-Hung Ng May 2007

On The Gauge Equivalence Of Twisted Quantum Doubles Of Elementary Abelian And Extra-Special 2-Groups, Christopher Goff, Geoffrey Mason, Siu-Hung Ng

Christopher Goff

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group H by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle w is gauge equivalent to a twisted …


An Explicit Fusion Algebra Isomorphism For Twisted Quantum Doubles Of Finite Groups, Christopher Goff Dec 2004

An Explicit Fusion Algebra Isomorphism For Twisted Quantum Doubles Of Finite Groups, Christopher Goff

Christopher Goff

We exhibit an isomorphism between the fusion algebra of the quantum double of an extraspecial p-group, where p is an odd prime, and the fusion algebra of a twisted quantum double of an elementary abelian group of the same order.


Bounding The Firing Synchronization Problem On A Ring, André Berthiaume, Todd Bittner, Ljubomir Perković, Amber Settle, Janos Simon Jun 2004

Bounding The Firing Synchronization Problem On A Ring, André Berthiaume, Todd Bittner, Ljubomir Perković, Amber Settle, Janos Simon

Amber Settle

In this paper we improve the upper and lower bounds on the complexity of solutions to the firing synchronization problem on a ring. In this variant of the firing synchronization problem the goal is to synchronize a ring of identical finite automata. Initially, all automata are in the same state except for one automaton that is designated as the initiator for the synchronization. The goal is to define the set of states and the transition function for the automata so that all machines enter a special fire state for the first time and simultaneously during the final round of the …


A Family Of Isomorphic Fusion Algebras Of Twisted Quantum Doubles Of Finite Groups, Christopher Goff Dec 2002

A Family Of Isomorphic Fusion Algebras Of Twisted Quantum Doubles Of Finite Groups, Christopher Goff

Christopher Goff

Let D<sup>ω</sup>(G) be the twisted quantum double of a finite group, G, where ω∈Z<sup>3</sup>(G,C∗). For each n∈N, there exists an ω such that D(G) and D<sup>ω</sup>(E) have isomorphic fusion algebras, where G is an extraspecial 2-group with 2<sup>2n+1</sup> elements, and E is an elementary abelian group with |E|=|G|.


Smaller Solutions For The Firing Squad, Amber Settle, Janos Simon Mar 2002

Smaller Solutions For The Firing Squad, Amber Settle, Janos Simon

Amber Settle

In this paper we improve the bounds on the complexity of solutions to the firing squad problem, also known as the firing synchronization problem. In the firing synchronization problem we consider a one-dimensional array of n identical finite automata. Initially all automata are in the same state except for one automaton designated as the initiator for the synchronization. Our results hold for the original problem, where the initiator may be located at either endpoint, and for the variant where any one of the automata may be the initiator, called the generalized problem. In both cases, the goal is to define …


Explicit Inversion Formulas For Toeplitz Band Matrices, William F. Trench Dec 1984

Explicit Inversion Formulas For Toeplitz Band Matrices, William F. Trench

William F. Trench

No abstract provided.


On Periodicities Of Certain Sequences Of Residues, William F. Trench Dec 1959

On Periodicities Of Certain Sequences Of Residues, William F. Trench

William F. Trench

No abstract provided.