Ridho Alfarisi

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 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).

Local Edge Antimagic Coloring of Comb Product of Graphs

All graph in this paper are finite, simple and connected graph. Let G(V, E) be a graph of vertex set V and edge set E. A bijection f : V (G) −→ {1, 2, 3, ..., |V (G)|} is called a local edge antimagic labeling if for any two adjacent edges e1 and e2, w(e1) 6= w(e2), where for e = uv ∈ G, w(e) = f(u) + f(v). Thus, any local edge antimagic labeling induces a proper edge coloring of G if each edge e is assigned the color w(e). The local edge antimagic hromatic number γlea(G) is the minimum number of colors taken over all colorings induced by local edge antimagic labelings of G. In this paper, we have found the lower bound of the local edge antimagic coloring of G . H and determine exact value local edge antimagic coloring of G . H.

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.