In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every Kn properly edge-coloured by n−1 colours has n/2 edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-c...

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ω(f) = ∑ v∈V |f(v)|. The 2-rainbow domination number of a graph G, denoted by γr2...

An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer k, an edge-colouring of a graph G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The rainbow k-connection number rck(G) is defined to be the minimum integer t such that there exists an edge-colouring of G with t colours which is rainbow ...

Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of ...

Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to bebegin{itemize}item a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The R...

Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the set of in-neighbors of v. The weight of a kRDF f is the value ω(f) = ∑ v∈V |f(v...

It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G − e) ≤ γ(G) + 1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number γw and the connected domination number γc, i.e., we show that γw(G) ≤ γw(G− e) ≤ γw(G) + 1 and γc(...

In this paper we consider the effect of edge contraction on the domination number and total domination number of a graph. We define the (total) domination contraction number of a graph as the minimum number of edges that must be contracted in order to decrease the (total) domination number. We show both of this two numbers are at most three for any graph. In view of this result, we classify gra...

Denote the total domination number of a graph G by γt(G). A graph G is said to be total domination edge critical, or simply γtcritical, if γt(G + e) < γt(G) for each edge e ∈ E(G). For 3t-critical graphs G, that is, γt-critical graphs with γt(G) = 3, the diameter of G is either 2 or 3. We characterise the 3t-critical graphs G with diam G = 3.

Edge Roman Star Domination Number on Graphs Angshu Kumar Sinha, Akul Rana and Anita Pal Department of Mathematics, NSHM Knowledge Campus Durgapur -713212, INDIA. e-mail: [email protected] Department of Mathematics, Narajole Raj College Narajole, Paschim Medinipur721211, INDIA. e-mail: [email protected] Department of Mathematics, National Institute of Technology Durgapur Durgapur-713209, I...