Cubic graphs with total domatic number at least two

Cubic graphs with total domatic number at least two

Let G be a graph. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

on the total domatic number of regular graphs

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

Total domination in cubic Knodel graphs

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

The Domatic Number Problem in Interval Graphs

The total {k}-domatic number of digraphs

For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V (D) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (D), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) consists of all vertices of D from which arcs go into v. A set {f1, f2, . . . , fd} of total {k}-dominating functions of D with the property that ∑ d i=1 fi(...