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Improving some bounds for dominating Cartesian products

2003
*
Discussiones Mathematicae Graph Theory
*

He

doi:10.7151/dmgt.1201
fatcat:6o4amed5vjc3llrid7ip67epze
*conjectured*that γ(G)γ(H) is a*lower**bound**for**the**domination**number**of**the*Cartesian product*of*any two graphs G*and*H. ...*For*example, we establish a new*lower**bound**for**the**domination**number**of*T T , when T is a tree,*and*we improve an*upper**bound**of*Vizing in*the*case when one*of**the*graphs has k > 1*dominating*sets which ...*Upper**Bounds*Nearly all*the*published results on*domination**of*Cartesian products have been motivated by*Vizing's**conjecture*,*and*so authors have been interested in*lower**bounds**for**the**domination**number*...##
###
Vizing's conjecture: a survey and recent results

2011
*
Journal of Graph Theory
*

*For*instance, several new properties

*of*a minimal counterexample

*to*

*the*

*conjecture*are obtained

*and*a

*lower*

*bound*

*for*

*the*

*domination*

*number*is proved

*for*products

*of*claw-free graphs with arbitrary graphs ...

*Vizing's*

*conjecture*from 1968 asserts that

*the*

*domination*

*number*

*of*

*the*Cartesian product

*of*two graphs is at least as large as

*the*product

*of*their

*domination*

*numbers*. ... Acknowledgements

*The*authors would like

*to*express their sincere appreciation

*to*

*the*Department

*of*Mathematics at Furman University. ...

##
###
Bounding the k-rainbow total domination number
[article]

2020
*
arXiv
*
pre-print

Recently

arXiv:2003.09470v1
fatcat:26lsvlxl6bak5muzutgqp345mi
*the*notion*of*k-rainbow total*domination*was introduced*for*a graph G, motivated by a desire*to*reduce*the*problem*of*computing*the*total*domination**number**of**the*generalized prism G K_k*to*an ... By stating a Vizing-like*conjecture**for*rainbow total*domination*we present a different viewpoint on*Vizing's*original*conjecture*in*the*case*of*bipartite graphs. ... J1-1692*and*bilateral projects between Slovenia*and*United states*of*America BI-US/18-20-061*and*BI-US/18-20-052. ...##
###
Bounds On (t,r) Broadcast Domination of n-Dimensional Grids
[article]

2022
*
arXiv
*
pre-print

Our main results provide some

arXiv:1908.07586v5
fatcat:5w5wgxkdqrfhtcicsymscswu2a
*upper**and**lower**bounds*on*the*density*of*a (t, r)*dominating*pattern*of*an infinite grid, as well as methods*of*computing them. ... Also, when r ≥ 2 we describe a family*of*counterexamples*to*a generalization*of**Vizing's**Conjecture**to*(t,r) broadcast*domination*. ...*The*(t, r) analog*of**Vizing's**Conjecture*Since (t, r) broadcast*domination*theory generalizes*domination*theory*and*distance*domination*theory, it is reasonable*to*ask whether*Vizing's**conjecture*applies ...##
###
The spectral radius of edge chromatic critical graphs

2016
*
Linear Algebra and its Applications
*

We obtain some

doi:10.1016/j.laa.2015.11.019
fatcat:gh27xwbbi5f6dgvqgvyenwlxmu
*lower**bounds**for*ρ(G)*and*µ(G),*and*present some cases where*the**conjectures*are true. Finally, several open problems are also proposed. ... In this paper, we consider two weaker versions*of**Vizing's**conjecture*, which concern*the*spectral radius ρ(G)*and**the*signless Laplacian spectral radius µ(G)*of*G. ...*The*authors are grateful*to**the*referee*for*their valuable comments which resulted in a significant improvement*of*this paper. Feng ...##
###
0011 | Domination Theory And Beyond
[article]

2022
*
Zenodo
*

Collections

doi:10.5281/zenodo.6320130
fatcat:hl556vg4j5hkhbgehemcanjzoy
*of*manuscripts about*domination**and*beyond. ...*The*author is highly thankful*to**the*Editor-in-Chief*and**the*referees*for*their valuable comments*and*suggestions*for*improving*the*paper. ...*The*author is highly thankful*to**the*Editor-in-Chief*and**the*referees*for*their valuable comments*and*suggestions*for*improving*the*paper. Acknowledgments We thank just about everybody. ...##
###
On outer-connected domination for graph products
[article]

2017
*
arXiv
*
pre-print

We investigate

arXiv:1708.00188v1
fatcat:kpyfwkpjy5bzrlucihg53pzne4
*the*existence*of*outer-connected*dominating*set in lexicographic product*and*Corona*of*two arbitrary graphs,*and*we present*upper**bounds**for*outer-connected*domination**number*in lexicographic ... Also, we establish an equivalent form*of**the**Vizing's**conjecture**for*outer-connected*domination**number*in lexicographic*and*Cartesian product as γ̃_̃c̃(G ∘ K)γ̃_̃c̃(H ∘ K) ≤γ̃_̃c̃(G H)∘ K. ... In 2010, Gasper Mekis [11] gave a*lower**bound**for**the**domination**number**of*a direct product*and*proved that this*bound*is sharp. ...##
###
Sketchy Tweets: Ten Minute Conjectures in Graph Theory

2012
*
The Mathematical intelligencer
*

ACKNOWLEDGMENTS We would like

doi:10.1007/s00283-012-9275-2
fatcat:lnaa2x6y7ze6nednpyzyi24lnu
*to*thank*the*anonymous referees*for*several suggestions that improved*the*quality*and*readability*of**the*paper. ... We must be content with*lower**and**upper**bounds*. An inductive argument gives RðnÞ 2n À 2 n À 1 . ...*For*example, a minimal counterexample*to**Vizing's**conjecture*must have*domination**number*larger than 3; adding an edge between two nonadjacent vertices decreases*the**domination**number*;*and*every vertex ...##
###
Domination Game and an Imagination Strategy

2010
*
SIAM Journal on Discrete Mathematics
*

A connection with

doi:10.1137/100786800
fatcat:m32ng4ihgrbnpbnp5vamjwhmxu
*Vizing's**conjecture*is established,*and*a*lower**bound*on*the** ...*The*game*domination**number*γ g (G) is*the**number**of*vertices chosen when*Dominator*starts*the*game*and**the*Staller-start game*domination**number*γ g (G) when Staller starts*the*game. ... We don't know whether*the**upper**bound**of*Theorem 6 can be*lowered*by 1*and*hence pose:*Conjecture*1 Pairs (k, k + 2)*for*k ≥ 1 are not realizable. ...##
###
Rainbow Turán Methods for Trees
[article]

2022
*
arXiv
*
pre-print

We explore

arXiv:2203.13765v1
fatcat:ne5gs7dsbrbxhlhxoqfqb7jwpa
*the*reduction method*for*finding*upper**bounds*on rainbow Turán*numbers*,*and*use this*to*inform results*for**the*rainbow Turán*numbers**of*double stars, caterpillars,*and*perfect binary trees. ...*The*rainbow Turán*number**of*a graph H, ex^*(n,H), is*the*largest*number**of*edges*for*an n vertex graph G which can be properly edge colored with no rainbow H subgraph. ... Therefore we have*the*following*upper**bound*: ex ⋆ (n, DS r,s ) ≤ ex(n, DS r,s+r ) ≤ (s+2r)n 2 .*The**lower**bound*is shown by a simple*application**of**Vizing's*theorem, as in Theorem 4.5. ...##
###
Page 42 of Mathematical Reviews Vol. , Issue 95a
[page]

1995
*
Mathematical Reviews
*

*The*author presents an

*upper*

*bound*on

*the*size

*of*a graph with a given

*domination*

*number*, which refines

*the*

*bound*by V. G. Vizing [Dokl. Akad. ...

*The*authors obtain a

*lower*

*bound*on

*the*size

*of*a maximum matching in a reduced graph. As a consequence, they verify

*and*strengthen a

*conjecture*

*of*A. Benhocine et al. [J. ...

##
###
Upper $k$-tuple total domination in graphs

2017
*
Pure and Applied Mathematics Quarterly
*

Also, we find some results on

doi:10.4310/pamq.2017.v13.n4.a1
fatcat:mw37ofzp2ngshjlrtisxrdrlxm
*the**upper*k-tuple total*domination**number**of**the*Cartesian*and*cross product graphs. ... In this paper, we introduce*the*concept*of**upper*k-tuple total*domination**number**of*G as*the*maximum cardinality*of*a minimal k-tuple total*dominating*set*of*G,*and*study*the*problem*of*finding a minimal ... Over more than fifty years (see [1]*and*references therein),*Vizing's**Conjecture*has been shown*to*hold*for*certain restricted classes*of*graphs,*and*furthermore,*upper**and**lower**bounds*on*the*inequality ...##
###
Towards the linear arboricity conjecture
[article]

2018
*
arXiv
*
pre-print

This

arXiv:1809.04716v1
fatcat:qicsfki6evau3mowoumimzsvgq
*conjectured**upper**bound*would be best possible, as is easily seen by taking G*to*be a regular graph. ... A famous*conjecture*due*to*Akiyama, Exoo,*and*Harary from 1980 asserts that la(G)≤ (Δ(G)+1)/2 , where Δ(G) denotes*the*maximum degree*of*G. ... Probabilistic estimates Throughout this paper, we will make extensive use*of**the*following well-known*bound*on*the**upper**and**lower*tails*of*a sum*of*independent indicators, due*to*Chernoff (see, e.g., ...##
###
Domination analysis of some heuristics for the traveling salesman problem

2002
*
Discrete Applied Mathematics
*

However, in this case

doi:10.1016/s0166-218x(01)00268-2
fatcat:gbmg3p34hbbffkuwi2lpiacx4m
*the*guaranteed*lower**bound*on*the**domination**number*would be half that*of**the*corresponding ATSP algorithm. ? ... A special case*of*this algorithm is shown*to*have complexity O(n 2 )*and**domination**number*at least n−2 k=0 (k!). ... Acknowledgements We are thankful*to**the*referees*for*their suggestions which improved*the*presentation*of**the*paper. ...##
###
Integer domination of Cartesian product graphs

2015
*
Discrete Mathematics
*

In this paper, we utilize

doi:10.1016/j.disc.2015.01.032
fatcat:5vkv2bnqqrdapevqizuod5wrgi
*the*approach developed by Clark*and*Suen (2000)*and*properties*of*binary matrices*to*prove a "Vizing-like" inequality on minimum {k}-*dominating*multisets*of*graphs G, H*and**the*... Specifically, denoting*the*size*of*a minimum {k}-*dominating*multiset as γ {k} (G), we demonstrate that γ {k} (G)γ {k} (H) ≤ 2k γ {k} (G H) . ... Acknowledgements*The*authors would like*to*acknowledge*the*support*of*NSF-CMMI-0926618,*the*Rice University VIGRE program (NSF DMS-0739420*and*EMSW21-VIGRE),*and**the*Global Initiatives ...
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