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Improving some bounds for dominating Cartesian products

Bert L. Hartnell, Douglas F. Rall
2003 Discussiones Mathematicae Graph Theory  
He 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  ... 
doi:10.7151/dmgt.1201 fatcat:6o4amed5vjc3llrid7ip67epze

Vizing's conjecture: a survey and recent results

Boštjan Brešar, Paul Dorbec, Wayne Goddard, Bert L. Hartnell, Michael A. Henning, Sandi Klavžar, and Douglas F. Rall
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.  ... 
doi:10.1002/jgt.20565 fatcat:toqekygjjrbqpjcpbswc6grxki

Bounding the k-rainbow total domination number [article]

Kerry Ojakian, Riste Skrekovski, Aleksandra Tepeh
2020 arXiv   pre-print
Recently 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.  ... 
arXiv:2003.09470v1 fatcat:26lsvlxl6bak5muzutgqp345mi

Bounds On (t,r) Broadcast Domination of n-Dimensional Grids [article]

Tom Shlomi
2022 arXiv   pre-print
Our main results provide some 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  ... 
arXiv:1908.07586v5 fatcat:5w5wgxkdqrfhtcicsymscswu2a

The spectral radius of edge chromatic critical graphs

Lihua Feng, Jianxiang Cao, Weijun Liu, Shifeng Ding, Henry Liu
2016 Linear Algebra and its Applications  
We obtain some 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  ... 
doi:10.1016/j.laa.2015.11.019 fatcat:gh27xwbbi5f6dgvqgvyenwlxmu

0011 | Domination Theory And Beyond [article]

Henry Garrett
2022 Zenodo  
Collections 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.  ... 
doi:10.5281/zenodo.6320130 fatcat:hl556vg4j5hkhbgehemcanjzoy

On outer-connected domination for graph products [article]

M. Hashemipour, M. R. Hooshmandasl, A. Shakiba
2017 arXiv   pre-print
We investigate 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.  ... 
arXiv:1708.00188v1 fatcat:kpyfwkpjy5bzrlucihg53pzne4

Sketchy Tweets: Ten Minute Conjectures in Graph Theory

Anthony Bonato, Richard J. Nowakowski
2012 The Mathematical intelligencer  
ACKNOWLEDGMENTS We would like 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  ... 
doi:10.1007/s00283-012-9275-2 fatcat:lnaa2x6y7ze6nednpyzyi24lnu

Domination Game and an Imagination Strategy

Boštjan Brešar, Sandi Klavžar, Douglas F. Rall
2010 SIAM Journal on Discrete Mathematics  
A connection with 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.  ... 
doi:10.1137/100786800 fatcat:m32ng4ihgrbnpbnp5vamjwhmxu

Rainbow Turán Methods for Trees [article]

Vic Bednar, Neal Bushaw
2022 arXiv   pre-print
We explore 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.  ... 
arXiv:2203.13765v1 fatcat:ne5gs7dsbrbxhlhxoqfqb7jwpa

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

Adel P. Kazemi
2017 Pure and Applied Mathematics Quarterly  
Also, we find some results on 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  ... 
doi:10.4310/pamq.2017.v13.n4.a1 fatcat:mw37ofzp2ngshjlrtisxrdrlxm

Towards the linear arboricity conjecture [article]

Asaf Ferber, Jacob Fox, Vishesh Jain
2018 arXiv   pre-print
This 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.,  ... 
arXiv:1809.04716v1 fatcat:qicsfki6evau3mowoumimzsvgq

Domination analysis of some heuristics for the traveling salesman problem

Abraham Punnen, Santosh Kabadi
2002 Discrete Applied Mathematics  
However, in this case 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.  ... 
doi:10.1016/s0166-218x(01)00268-2 fatcat:gbmg3p34hbbffkuwi2lpiacx4m

Integer domination of Cartesian product graphs

K. Choudhary, S. Margulies, I.V. Hicks
2015 Discrete Mathematics  
In this paper, we utilize 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  ... 
doi:10.1016/j.disc.2015.01.032 fatcat:5vkv2bnqqrdapevqizuod5wrgi
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