On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of the finite poset

Bouchemakh, Isma; Ouatiki, Saliha

doi:10.1016/j.endm.2005.06.057

Cited by 2 publications

(2 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Chen et al, [8] shown that every r-regular graph has d t (G) ≥ (1 − o(1))r/ log r as r → ∞, and the proportion of r-regular graphs for which d t (G) ≤ (1 + o(1))r/ log r tends to 1 as |V (G)| → ∞. Bouchemakh and Ouatiki [6] studied the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of a finite poset. In [21], Koivisto et al, showed that it is NP-complete to decide whether d t (G) ≥ 3 where G is a bipartite planar graph of bounded maximum degree.…”

Section: Introductionmentioning

confidence: 99%

Disjoint Total Dominating Sets in Near-Triangulations

Francis¹,

Illickan²,

Jose³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Disjoint Total Dominating Sets in Near-Triangulations

Francis¹,

Illickan²,

Jose³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…See for instance, [6,13,20,21,23] and a survey of selected topics by Henning [19]. The concept of domatic number and total domatic number was introduced by Cockayne et al, in [10] and [9] respectively, and investigated further in [1,2,5,8,12,18,24,26,32,33]. In [33], Zelinka have shown the existence of graphs with very large minimum degree have a total domatic number 1.…”

Section: Introductionmentioning

confidence: 99%

On Domatic and Total Domatic Numbers of Product Graphs

Francis¹,

Rajendraprasad²

2021

Preprint

View full text Add to dashboard Cite

A domatic (total domatic) k-coloring of a 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 (neighborhood). The domatic (total domatic) number of G, denoted d(G) (d t (G)), is the maximum k for which G has a domatic (total domatic) k-coloring. In this paper, we show that for two non-trivial graphs G and H, the domatic and total domatic numbers of their Cartesian product G H is bounded above by max{|V (G)|, |V (H)|} and below by max{d(G), d(H)}. Both these bounds are tight for an infinite family of graphs. Further, we show that if H is bipartite, then d t (G H) is bounded below by 2 min{d t (G), d t (H)} and d(G H) is bounded below by 2 min{d(G), d t (H)}. These bounds give easy proofs for many of the known bounds on the domatic and total domatic numbers of hypercubes [8,31] and the domination and total domination numbers of hypercubes [16,22] and also give new bounds for Hamming graphs. We also obtain the domatic (total domatic) number and domination (total domination) number of ndimensional torus n i=1 C k i with some suitable conditions to each k i , which turns out to be a generalization of a result due to Gravier [13] and give easy proof of a result due to Klavžar and Seifter [23].

show abstract

On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of the finite poset

Cited by 2 publications

References 7 publications

Disjoint Total Dominating Sets in Near-Triangulations

Disjoint Total Dominating Sets in Near-Triangulations

On Domatic and Total Domatic Numbers of Product Graphs

Contact Info

Product

Resources

About