Regular totally domatically full graphs
type: Item
instance of: scholarly article
title: Regular totally domatically full graphs
zbMATH Open document ID: 0745.05065
DOI: 10.1016/0012-365X(90)90350-Q
author: Bohdan Zelinka
published in: Discrete Mathematics
publication date: 1992-06-25T00:00:00Z
review text: The total domatic number \(d_ t(G)\) of a graph \(G\) was studied by \textit{R. B. Allan, R. Laskar} and \textit{St. Hedetniemi} [Discrete Math. 49, 7-13 (1984; Zbl 0576.05028)]. Let \(md(G)\) be the minimum degree of a vertex in \(G\). A graph \(G\) is said to be totally domatically full if \(d_ t(G)=md(G)\). By a uniquely total domatic graph the author means a graph \(G\) in which exists exactly one total domatic partition with \(d_ t(G)\) classes. First, he considers bipartite graphs. Among other things he shows that no bipartite graph \(G\) with \(d_ t(G)\geq 2\) is uniquely totally domatic. A weaker concept called quasi-uniquely totally domatic graph is introduced and turns out to be useful in the study of directed graphs. Further the parameter \(d_ t(G)\) is transferred to digraphs. For instance it is shown that for positive integers \(k,n\), \(n\geq 3k\), there exists a tournament \(T\) of order \(n\) such that \(d_ t(T)=k\).
reviewed by: J.Sedláček
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 8140
zbMATH Keywords: totally domatically full graphs
zbMATH Keywords: domatic number
zbMATH Keywords: bipartite graphs
zbMATH Keywords: directed graphs
zbMATH Keywords: tournament
MaRDI profile type: Publication
cites work: A note on total domination
cites work: Q3941433
cites work: Total domination in graphs
cites work: Towards a theory of domination in graphs
cites work: Q3737457
Wikidata QID: Q127741531
description: scientific article; zbMATH DE number 8140