Saladin Uttunggadewa

The metric dimension of the lexicographic product of graphs

Abstract A set of vertices W resolves a graph G if every vertex is uniquely determined by its coordinate of distances to the vertices in W . The minimum cardinality of a resolving set of G is called the metric dimension of G . In this paper, we consider a graph which is obtained by the lexicographic product between two graphs. The lexicographic product of graphs G and H , which is denoted by G ∘ H , is the graph with vertex set V ( G ) × V ( H ) = { ( a , v ) | a ∈ V ( G ) , v ∈ V ( H ) } , where ( a , v ) is adjacent to ( b , w ) whenever a b ∈ E ( G ) , or a = b and v w ∈ E ( H ) . We give the general bounds of the metric dimension of a lexicographic product of any connected graph G and an arbitrary graph H . We also show that the bounds are sharp.

An upper bound for the ramsey number of a cycle of length four versus wheels

For given graphs G and H, the Ramsey numberR(G,H) is the smallest positive integer n such that every graph F of n vertices satisfies the following property: either F contains G or the complement of F contains H. In this paper, we show that the Ramsey number

On Super Edge-Magic Strength and Deficiency of Graphs

R(C_4, W_m) \leq m + \lceil \frac{m}{3} \rceil +1

Restricted size Ramsey number for path of order three versus graph of order five

for m ≥ 6. AMS Subject Classifications: 05C55, 05D10.

The Partition Dimension of a Subdivision of a Complete Graph

A graph G is called super edge-magic if there exists a one-to-one mapping f from V (G ) *** E (G ) onto {1, 2, 3, *** , |V (G )| + |E (G )|} such that for each uv *** E (G ), f (u ) + f (uv ) + f (v ) = c (f ) is constant and all vertices of G receive all smallest labels. Such a mapping is called super edge-magic labeling of G . The super edge-magic strength of a graph G is defined as the minimum of all c (f ) where the minimum runs over all super edge-magic labelings of G . Since not all graphs are super edge-magic, we define, the super edge-magic deficiency of a graph G as either minimum n such that G *** nK 1 is a super edge-magic graph or + *** if there is no such n . In this paper, the bound of super edge-magic strength and the super edge-magic deficiency of some families of graphs are obtained.

On The Restricted Size Ramsey Number.

Let

Some conditions for the existence of ( d,k )-digraphs

G

Locating-chromatic number for a graph of two components

and

The partition dimension of subdivision of a graph

H

Restricted size Ramsey number for P3 versus small paths

be simple graphs. The Ramsey number for a pair of graph