A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph G is strongly rainbow-connected. The minimum number of colors needed to make G rainbow-connected is known as the rainbow connection number...

We consider a proper coloring of plane graph such that no face is rainbow, where rainbow if any two vertices on its boundary have distinct colors. Such said to be anti-rainbow. A quadrangulation G in which all faces are bounded by cycle length 4. In this paper, we show the number colors anti-rainbow does not exceed $$3\alpha (G)/2$$ , $$\alpha (G)$$ independence G. Moreover, minimum degree 3 or...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

In this paper, we provide an upper bound for the k-tuple domination number that generalises known upper bounds for the double and triple domination numbers. We prove that for any graph G, ×k(G) ln( − k + 2)+ ln(∑k−1 m=1(k −m)d̂m + )+ 1 − k + 2 n, where ×k(G) is the k-tuple domination number; is the minimal degree; d̂m is the m-degree of G; = 1 if k = 1 or 2 and =−d if k 3; d is the average degree...

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) ofG is the mini...

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...

Let γ(G) and γ2,2(G) denote the domination number and (2, 2)domination number of a graph G, respectively. In this paper, for any nontrivial tree T , we show that 2(γ(T )+1) 3 ≤ γ2,2(T ) ≤ 2γ(T ). Moreover, we characterize all the trees achieving the equalities.

This paper provides lower orientable k-step domination number and upper orientable k-step domination number of complete r-partite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete r-partite graphs.

The susceptibility of 2 strains of rainbow trout Oncorhynchus mykiss, 1 from North America (TL) and 1 from Germany (GR), to Myxobolus cerebralis (the cause of salmonid whirling disease) was assessed following exposure to the infectious stages (triactinomyxons). Two laboratory experiments were conducted with age-matched rainbow trout of each strain. At the beginning of the study, the 2 trout str...

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such ...