A three-valued function f defined on the vertices of a graph G = ( V, E), f : V 4 {-I. 0. I }, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every 1~ t V, ,f(N[o]) > 1, where N[c] consists of I: and every vertex adjacent to 1’. The weight of a minus dominating function is f(V) = c f(u), over all vertices L: t V. The m...

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of Kn, there is a rainbow path on (3/4− o(1))n vertices, improving on the previously best bound of (2n + 1)/3 from [?]. Similarly, a k-rainbow path in a proper...

Let G be a graph with integral edge weights. A function d : V (G) → Zp is called a nowhere 0 mod p domination function if each v ∈ V satisfies ( d(v) + ∑ u∈N(v) w(u, v)d(u) ) 6= 0 mod p, where w(u, v) denotes the weight of the edge (u, v) and N(v) is the neighborhood of v. The subset of vertices with d(v) 6= 0 is called a nowhere 0 mod p dominating set. It is known that every graph has a nowher...

A Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex x with f (x) = 0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V ) = ∑ x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer ...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is compatible with the correspondence principle of Rainbow Gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR space...

Using ray theory, the Möbius shift of the (p-1)-order rainbow angle for a particle having an elliptical cross section is obtained to first order in the ellipticity as a function of the tilt of the ellipse with respect to the propagation direction of the incoming rays. The result is then adapted to the geometry of scattering of light rays from the sun by a falling water drop as a function of sun...

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...

A Roman dominating function on a graph G = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value w(f) = ∑ v∈V f(v). The Roman domination number of a graph G, denoted by γR(G), equals the minimum weight of a Roman dominating function on ...