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On Domatic and Total Domatic Numbers of Cartesian Products of Graphs
Bulletin of the Malaysian Mathematical Sciences Society, 2023A domatic \(k\)-coloring of a simple, undirected graph \(G\) is an assignment of \(k\) colors to the vertices of \(G\) such that each vertex contains vertices of all \(k\) colors in its closed neighborhood. The domatic number of \(G\), denoted \(d(G)\), is the maximum \(k\) for which \(G\) has a domatic \(k\)-coloring. A total domatic \(k\)-coloring is
P. Francis, Deepak Rajendraprasad
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Approximating the domatic number
Proceedings of the thirty-second annual ACM symposium on Theory of computing, 2000A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, δ the minimum degree, and ∆ the maximum degree.
Uriel Feige +2 more
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2023
Summary: The domatic coloring number \(d(G)\) of a graph \(G\) is the maximum \(k\) for which \(V(G)\) can be partitioned into \(k\) pairwise disjoint dominating sets. In this paper, we determined the domatic numbers of complete graphs, complete \(k\)-partite graphs, Johnson graphs \(J(n, 2)\), unicyclic graphs, bicyclic graphs and generalized \(\Theta\
Borzooei, Rajab Ali +1 more
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Summary: The domatic coloring number \(d(G)\) of a graph \(G\) is the maximum \(k\) for which \(V(G)\) can be partitioned into \(k\) pairwise disjoint dominating sets. In this paper, we determined the domatic numbers of complete graphs, complete \(k\)-partite graphs, Johnson graphs \(J(n, 2)\), unicyclic graphs, bicyclic graphs and generalized \(\Theta\
Borzooei, Rajab Ali +1 more
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Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph
2020The fractional domatic number of a graph G is the maximum ratio \(|\mathcal {F}|/m(\mathcal {F})\) over all families \(\mathcal {F}\) of dominating sets of G, where \(m(\mathcal {F})\) denotes the maximum number of times any particular vertex appears in \(\mathcal {F}\).
Wayne Goddard, Michael A. Henning
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VERTEX COVERING TRANSVERSAL DOMATIC NUMBER OF GRAPHS
Advances and Applications in Discrete Mathematics, 2017Summary: A dominating set \(S \subseteq V\) in a simple graph \(G=(V,E)\) is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of \(G\). The vertex covering transversal domination number \(\gamma_{\mathrm{vct}}(G)\) is the minimum cardinality among all vertex covering transversal dominating sets ...
Vasanthi, R., Subramanian, K.
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On the Roman {2}-domatic number of graphs
Discrete Mathematics, Algorithms and Applications, 2020Let [Formula: see text] be a graph. A Roman[Formula: see text]-dominating function [Formula: see text] has the property that for every vertex [Formula: see text] with [Formula: see text], either [Formula: see text] is adjacent to a vertex assigned [Formula: see text] under [Formula: see text], or [Formula: see text] is adjacent to at least two ...
A. Giahtazeh +2 more
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The k-domatic number of a graph
Czechoslovak Mathematical Journal, 2009zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Kämmerling, Karsten, Volkmann, Lutz
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Graphs with small Italian domatic number
Discrete Applied Mathematics, 2023zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Keith Gallegos, Jeremy Lyle
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The {k}-domatic number of a graph
Aequationes mathematicae, 2011zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Meierling, D. +2 more
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Approximating theDomatic Number
SIAM Journal on Computing, 2002Summary: A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let \(n\) denote the number of vertices, \(\delta\) the minimum degree, and \(\Delta\) the maximum degree. We
Feige, Uriel +3 more
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