SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS
SM Sheikholeslami, L Volkmann - Discrete Mathematics, Algorithms …, 2012 - World Scientific
SM Sheikholeslami, L Volkmann
Discrete Mathematics, Algorithms and Applications, 2012•World ScientificLet k be a positive integer, and let G be a simple graph with vertex set V (G). A function f: V
(G)→{±1,±2,…,±k} is called a signed total {k}-dominating function if∑ u∈ N (v) f (u)≥ k for
each vertex v∈ V (G). A set {f1, f2,…, fd} of signed total {k}-dominating functions on G with
the property that for each v∈ V (G), is called a signed total {k}-dominating family (of
functions) on G. The maximum number of functions in a signed total {k}-dominating family on
G is the signed total {k}-domatic number of G, denoted by. Note that is the classical signed …
(G)→{±1,±2,…,±k} is called a signed total {k}-dominating function if∑ u∈ N (v) f (u)≥ k for
each vertex v∈ V (G). A set {f1, f2,…, fd} of signed total {k}-dominating functions on G with
the property that for each v∈ V (G), is called a signed total {k}-dominating family (of
functions) on G. The maximum number of functions in a signed total {k}-dominating family on
G is the signed total {k}-domatic number of G, denoted by. Note that is the classical signed …
Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) → {±1, ±2, …, ±k} is called a signed total {k}-dominating function if ∑u∈N(v) f(u) ≥ k for each vertex v ∈ V(G). A set {f1, f2, …, fd} of signed total {k}-dominating functions on G with the property that for each v∈V(G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by . Note that is the classical signed total domatic number dS(G). In this paper, we initiate the study of signed total k-domatic numbers in graphs, and we present some sharp upper bounds for . In addition, we determine for several classes of graphs. Some of our results are extensions of known properties of the signed total domatic number.
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