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### On a Vizing-like conjecture for direct product graphs

Sandi Klavẑar, Blaẑ Zmazek
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let 7(G) be the domination number of a graph G, and let G ×H be the direct product of graphs G and H. It is shown that for any k t> 0 there exists a graph G such that 7(G × G) ~< 7(G) 2 -k.  ...  This in particular disproves a conjecture from [5] .  ...  Gravier and Khelladi [5] posed the following Vizing-like conjecture for the direct product: ?(G x H)/> ?(G)?(H). Here we show that for any k/> 0 there exists a graph G such that 7(G× G) ~< 7(G)2-k.  ...
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### Vizing-type bounds for graphs with induced subgraph restrictions [article]

Elliot Krop, Pritul Patel, Gaspar Porta
<span title="2017-05-14">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For any graphs G and H, we say that a bound is of Vizing-type if γ(G H)≥ c γ(G)γ(H) for some constant c. We show several bounds of Vizing-type for graphs G with forbidden induced subgraphs.  ...  In particular, if G is a triangle and K_1,r-free graph, then for any graph H, γ(G H)≥r/2r-1γ(G)γ(H).  ...  Since Vizing's conjecture is known for graphs with domination number one, two [7] , and three [6] , it follows that a graph G that is K 3 -free and P 5 -free satisfies Vizings Conjecture.  ...
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### On Vizing's conjecture

Boštjan Brešar
<span title="">2001</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
For the Cartesian product G2H Vizing's conjecture [10] states that γ(G2H) ≥ γ(G)γ(H) for every pair of graphs G, H.  ...  A dominating set D for a graph G is a subset of V (G) such that any vertex in V (G)−D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G.  ...  Acknowledgement I wish to thank Sandi Klavžar for helpful remarks.  ...
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### Author index to volume 156 (1996)

<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Zmazek, On a Vizing-like conjecture for direct product graphs (1 3) 243 246 Kreher, D.L., G.F. Royle and W.D. Wallis, A family of resolvable regular graph designs (Note) , J.H., S.-G. Kim and J.  ...  Jimbo, A series of identities for the coefficients of inverse matrices on a Hamming scheme (Note) H., Hamiltonian circuits in chordal bipartite graphs (Note) E., The structure of imperfect critically strongly-imperfect  ...
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### On the total {k}-domination number of Cartesian products of graphs

Ning Li, Xinmin Hou
<span title="2008-02-13">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tsm6lnbvf5ahzjp7czojbjig5i" style="color: black;">Journal of combinatorial optimization</a> </i> &nbsp;
Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63-69, 2005).  ...  In this paper, we show that for any graphs G and H without isolated vertices, As a corollary of this result, we have γ t (G)γ t (H ) ≤ 2γ t (G H ) for all graphs G and H without isolated vertices, which  ...  Vizing (1968) conjectured that: For any graphs G and H , γ (G)γ (H ) ≤ γ (G H ). The best general upper bound to date on γ (G)γ (H ) in terms of γ (G H ) is due to Clark and Suen (2000) .  ...
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### Upper \$k\$-tuple total domination in graphs

<span title="">2017</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zqclzy5yebar3oqqmppyjia4le" style="color: black;">Pure and Applied Mathematics Quarterly</a> </i> &nbsp;
Also, we find some results on the upper k-tuple total domination number of the Cartesian and cross product graphs.  ...  Let G = (V, E) be a simple graph. For any integer k ≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set.  ...  Also we present a Vizing-like conjecture on the upper k-tuple total domination number, and prove it for a family of graphs.  ...
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### Graph products and integer domination

Niluk John, Stephen Suen
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We consider the {k}-domination number γ {k} (G) of a graph G and the Cartesian product G H and the strong direct product G H of graphs G and H.  ...  We extend a result from Brešar et al. on γ (G H) for claw-free graphs G.  ...  Acknowledgments The authors would like to thank the referees for their careful reading of the manuscript and for their helpful comments. References  ...
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### Page 2773 of Mathematical Reviews Vol. , Issue 97E [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
; Maribor) On a Vizing-like conjecture for direct product graphs.  ...  Summary: “Let y(G) be the domination number of a graph G, and let G x H_ be the direct product of graphs G and H. It is shown that for any k > 0 there exists a graph G such that »(G x G) < 7(G)? —k.  ...
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### Computing global offensive alliances in Cartesian product graphs [article]

Ismael G. Yero, Juan A. Rodríguez-Velázquez
<span title="2012-07-25">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.  ...  A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at least one neighbor in S.  ...  Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.  ...
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### Cartesian Product and Acyclic Edge Colouring [article]

Rahul Muthu, C.R. Subramanian
<span title="2015-08-06">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This extends a recent result of [15] where tight and constructive bounds on a'(G) were obtained for a class of grid-like graphs which can be expressed as the cartesian product of a number of paths and  ...  We show that a'(G H)< a'(G) + a'(H) for any two graphs G and H such that max{a'(G), a'(H)} > 1. Here, G H denotes the cartesian product of G and H.  ...  It would be interesting to make a statement like a ′ (G✷H) ≤ a ′ (G) + ∆(H) for a wider class of graphs H, like the results obtained in [15] for grid-like graphs.  ...
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### Integer domination of Cartesian product graphs

K. Choudhary, S. Margulies, I.V. Hicks
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
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  ...  Cartesian product graph 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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### Computing global offensive alliances in Cartesian product graphs

Ismael G. Yero, Juan A. Rodríguez-Velázquez
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.  ...  A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at least one neighbor in S.  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2012.08.009">doi:10.1016/j.dam.2012.08.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m7hirrz5ffdcpaua5pkkxdrvga">fatcat:m7hirrz5ffdcpaua5pkkxdrvga</a> </span>
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### Total Colourings - A survey [article]

Geetha Jayabalan and Narayanan N and K Somasundaram
<span title="2018-12-14">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we give a survey on total coloring of graphs.  ...  Vizing and Behzed conjectured that the total coloring can be done using at most Δ(G)+2 colors, where Δ(G) is the maximum degree of G. It is not settled even for planar graphs.  ...  Katja Prnaver and Blaž Zmazek [71] verified the conjecture for direct product of a path and any graph G with χ ′ (G) = ∆(G).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1812.05833v1">arXiv:1812.05833v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2igatxvf55bcncqp2q2riwpp2y">fatcat:2igatxvf55bcncqp2q2riwpp2y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200824231055/https://arxiv.org/pdf/1812.05833v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8c/41/8c41ffb66ace775a5aaec5a39dac4d0f90f12456.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1812.05833v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### RELATIONSHIP BETWEEN EQUITABLE TOTAL COLORING AND RANGE COLORING IN SOME REGULAR GRAPHS

Abel Rodolfo García Lozano, Angelo Santos Siqueira, Clícia Valladares Peixoto Friedmann, Samuel Jurkiewicz Jurkiewicz
<span title="">2016</span> <i title="FapUNIFESP (SciELO)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3cu5b5xldjhcvatxke4awifjni" style="color: black;">Pesquisa Operacional</a> </i> &nbsp;
For this purpose, we use some relationships between equitable total coloring and range (vertex) coloring in some regular graphs.  ...  We also show that there are infinite graphs satisfying this theorem. Such graphs are called Harmonics. We generate Harmonic Graphs which are Cartesian products of cycles and their complements.  ...  Thus, the Harmonic Graphs have relevant characteristics for admitting a total equitable coloring that meets the Vizing-Behzad Conjecture, as demonstrated in Theorem 2.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1590/0101-7438.2016.036.01.0101">doi:10.1590/0101-7438.2016.036.01.0101</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xh26f256z5guxhxe647op2ueja">fatcat:xh26f256z5guxhxe647op2ueja</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727072527/http://www.scielo.br/pdf/pope/v36n1/1678-5142-pope-36-01-00101.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b1/6b/b16be81918d9487c622cf22f3bfafff3c0bb1bb8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1590/0101-7438.2016.036.01.0101"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> scielo.br </button> </a>

### 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
<span title="2011-02-09">2011</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
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.  ...  Its financial support from the Wylie Enrichment Fund enabled them to gather in a relaxed atmosphere to work on the conjecture.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.20565">doi:10.1002/jgt.20565</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/toqekygjjrbqpjcpbswc6grxki">fatcat:toqekygjjrbqpjcpbswc6grxki</a> </span>
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