Results 11 to 20 of about 113,121 (99)
Discussiones Mathematicae Graph Theory, 2001 Summary: 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 \(\gamma(G)\) is the size of a minimum dominating set for \(G\). For the Cartesian product \(G\square H\) Vizing's conjecture [cf. \textit{V. G. Vizing}, Vychisl.
Bresar, Bostjan
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An Optimization-Based Sum-of-Squares Approach to Vizing's Conjecture [PDF]
Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation, 2019 Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question.
Elisabeth Gaar +3 more
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An Algebraic Exploration of Dominating Sets and Vizing's Conjecture [PDF]
The Electronic Journal of Combinatorics, 2012 Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the ...
Susan Margulies, Illya V. Hicks
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Vizing'S Weaker Conjecture [PDF]
, 2017 Vizing conjectured that G is a simple and ∆-critical graph with m edges then . In this paper we prove the conjecture for graphs with and .
M. Santhi, P. Anitha
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Vizing's edge-recoloring conjecture holds.
Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications, 2023 In 1964 Vizing proved that starting from any $k$-edge-coloring of a graph $G$ one can reach, using only Kempe swaps, a $(\Delta+1)$-edge-coloring of $G$ where $\Delta$ is the maximum degree of $G$. One year later he conjectured that one can also reach a $\Delta$-edge-coloring of $G$ if there exists one. Bonamy \textit{et. al} proved that the conjecture
Narboni, Jonathan
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Sum-of-squares certificates for Vizing's conjecture via determining Gröbner bases [PDF]
Journal of Symbolic Computation, 2023 The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs $G$ and $H$ is at least the product of the domination numbers of $G$ and $H$. Recently Gaar, Krenn, Margulies and Wiegele used the graph class $\mathcal{G}$ of all graphs with $n_\mathcal{G}$ vertices and domination number $k_\mathcal{G ...
Elisabeth Gaar, Melanie Siebenhofer
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On a strong form of Oliver’s p-group conjecture. [PDF]
, 2011 We introduce a stronger and more tractable form of Olivers p-group conjecture, and derive a reformulation in terms of the modular representation theory of a quotient group.
Green, David J. +7 more
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Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares [PDF]
, 2021 Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question.
Margulies, Susan +3 more
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Domination in graphs: Vizing's conjecture [PDF]
, 2023 Vizing's conjecture remains one of the biggest open problems in domination in graph theory today. The conjecture states that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the
Harnaker, Zahraa
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An Improved Bound in Vizing’s Conjecture [PDF]
Graphs and Combinatorics, 2019 A well-known conjecture of Vizing is that $γ(G \square H) \ge γ(G)γ(H)$ for any pair of graphs $G, H$, where $γ$ is the domination number and $G \square H$ is the Cartesian product of $G$ and $H$. Suen and Tarr, improving a result of Clark and Suen, showed $γ(G \square H) \ge \frac{1}{2}γ(G)γ(H) + \frac{1}{2}\min(γ(G),γ(H))$.
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