nan Dafik

The non-isolated resolving number of k-corona product of graphs

Let all graphs be a connected and simple graph. A set W = {w1, w2, w3, . . . , wk} of veretx set of G, the k−vector ordered r(v|W ) = (d(x,w1), d(x,w2), . . . , d(x,wk)) of is a representation of v with respect to W , for d(x,w) is the distance between the vertices x and w. The set W is called a resolving set for G if different vertices of G have distinct representation. The metric dimension is the minimum cardinality of resolving set W , denoted by dim(G). Through analogue, the resolving set W of G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. The non-isolated resolving number is the minimum cardinality of non-isolated resolving set W , denoted by nr(G). In our paper, we determine the non isolated resolving number of k-corona product graph.

On Rainbow k-Connection Number of Special Graphs and It’s Sharp Lower Bound

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c : E(G) → {1, 2, ..., s}, s ∈ N. A path of 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 G is said to be a rainbow connected graph if there exists a rainbow u − v path for every two vertices u and v of G. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of k colors required to edge color the graph such that the graph is rainbow connected. Furthermore, for an l-connected graph G and an integer k with 1 ≤ k ≤ l, the rainbow k-connection number rck (G) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. In this paper, we determine the exact values of rainbow connection number of some special graphs and obtain a sharp lower bound.

On super H−antimagicness of an edge comb product of graphs with subgraph as a terminal of its amalgamation

All graphs in this paper are simple, finite, and undirected graph. Let r be a edges of H. The edge comb product between L and H, denoted by LH, is a graph obtained by taking one copy of L and |E(L)| copies of H and grafting the i-th copy of H at the edges r to the i-th edges of L, we call such a graph as an edge comb product of graph with subgraph as a terminal of its amalgamation, denoted by G = KAmal(H, L ⊂ H, n). The graph G is said to admits an (a, d)-H-antimagic total labeling if there exist a bijection f : V(G) E(G) → {1, 2, ..., |V (G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights W (H) = ∑ v∈V(H) f(v) + ∑e∈E(H) f(e) form an arithmetic sequence {a, a + d, a + 2d, ..., a + (t − 1)d}, where a and d are positive integers and t is the number of all subgraphs isomorphic to H. An (a, d)-H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we will study the super H−antimagicness of disjoint union of edge comb product of graphs with subgraph as a terminal of its amalgamation.

Bound of Distance Domination Number of Graph and Edge Comb Product Graph

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. For an integer 1 ≤ k ≤ diam(G), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G)\S is at distance at most k from some vertex of S. The k-domination number, denoted by γ k (G), of G is the minimum cardinality of a k-dominating set of G. In this paper, we improve the lower bound on the distance domination number of G regarding to the diameter and minimum degree as well as the upper bound regarding to the order and minimum k distance neighbourhood. In addition, we determine the bound of distance domination number of edge comb product graph.

On the rainbow coloring for some graph operations

Let G = (V, E) be a nontrivial, finite, simple and undirected connected graph on which is defined a coloring f : E(G) → {1,2, …, k}, k ∈ N. The adjacent edges may be colored the same colors. 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 G is rainbow connected if there exists a rainbow u – v path for every two vertices u and v of G. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of k colors required to edge color the graph such that the graph is rainbow connected. In this paper, we determine the exact values of rainbow connection number of some special graph operations, namely cartesian product, tensor product, composition of two special graphs and also amalgamation of special graphs. The result shows that all exact values of rc(G) attain a lower bound of the rainbow connectivity, namely diam(G).

On the r-dynamic chromatic number of the corronation by complete graph

In this paper we will study the r-dynamic chromatic number of the coronation by complete graph. A proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. The r-dynamic chromatic number, χ r (G) is the minimum k such that graph G has an r-dynamic k-coloring. We will obtain lower bound of the r-dynamic chromatic number of , and We also study the exact value of the r-dynamic chromatic number of and for m, n > 3.

On the locating domination number of P n [trianglerightequal] H graph

A subset D of V(G) is called a vertex dominating set of G if every vertex not in D is adjacent to some vertices in D. A graph G = (V, E) is called a locating dominating set if for every two vertices . Locating dominating number is the minimum cardinality of a locating dominating set. The value of locating domination number is . Edge comb product denoted by is a graph obtained by taking one copy of G and |E(G)| copies of H and grafting the i-th copy of H at the edge e to the i-th edge of G. This paper studies about locating domination number in edge comb product graph where G is path graph Pn and H is complete graph Km , star graph Sm , triangular book graph BTm , path graph Pm , friendship graph and fan graph Fm .

On the star partition dimension of comb product of cycle and complete graph

Let G = (V, E) be a connected graphs with vertex set V (G), edge set E(G) and S ⊆ V (G). For an ordered partition Π = {S 1, S 2, S 3, ..., Sk } of V (G), the representation of a vertex v ∈ V (G) with respect to Π is the k-vectors r(v|Π) = (d(v, S 1), d(v, S 2), ..., d(v, Sk )), where d(v, Sk ) represents the distance between the vertex v and the set Sk , defined by d(v, Sk ) = min{d(v, x)|x ∈ Sk}. The partition Π of V (G) is a resolving partition if the k-vektors r(v|Π), v ∈ V (G) are distinct. The minimum resolving partition Π is a partition dimension of G, denoted by pd(G). The resolving partition Π = {S 1, S 2, S 3, ..., Sk} is called a star resolving partition for G if it is a resolving partition and each subgraph induced by Si , 1 ≤ i ≤ k, is a star. The minimum k for which there exists a star resolving partition of V (G) is the star partition dimension of G, denoted by spd(G). Finding a star partition dimension of G is classified to be a NP-Hard problem. Furthermore, the comb product between G and H, denoted by G H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. By definition of comb product, we can say that V (G H) = {(a, u)|a ∈ V (G), u ∈ V (H)} and (a, u)(b, v) ∈ E(G H) whenever a = b and uv ∈ E(H), or ab ∈ E(G) and u = v = o. In this paper, we will study the star partition dimension of comb product of cycle and complete graph, namely Cn Km and Km Cn for n ≥ 3 and m ≥ 3.