Hilda Assiyatun

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.

3-star factors in random d -regular graphs

The small subgraph conditioning method first appeared when Robinson and the second author showed the almost sure hamiltonicity of random d-regular graphs. Since then it has been used to study the almost sure existence of, and the asymptotic distribution of, regular spanning subgraphs of various types in random d-regular graphs and hypergraphs. In this paper, we use the method to prove the almost sure existence of 3-star factors in random d-regular graphs. This is essentially the first application of the method to nonregular subgraphs in such graphs.

On Ramsey (3K2, K3) – minimal graphs

The Ramsey graph theory has many interesting applications, such as in the fields of communications, information retrieval, and decision making. One of growing topics in Ramsey theory is Ramsey minimal graph. For any given graphs G and H, find graphs F such that any red-blue coloring of all edges of F contains either a red copy of G or a blue copy of H. If this condition is not satisfied by the graph F − e, then we call the graph F as a Ramsey (G, H) – minimal. In this paper, we derive the properties of (3K2, K3) – minimal graphs. We, then, characterize all Ramsey (3K2, K3) – minimal graphs.

Maximum induced matchings of random regular graphs

An induced matching of a graph G = (V,E) is a matching

Total Irregularity Strength of Three Families of Graphs

{\mathcal M}

The complete list of Ramsey

such that no two edges of

(2K_2,K_4)

{\mathcal M}

-minimal graphs

are joined by an edge of E/

The Ramsey numbers for disjoint unions of graphs

{\mathcal M}

On Unicyclic Ramsey (mK2, P3)−Minimal Graphs☆

In general, the problem of finding a maximum induced matching of a graph is known to be NP-hard. In random d-regular graphs, the problem of finding a maximum induced matching has been studied for d ∈ {3, 4, ..., 10 }. This was due to Duckworth et al.(2002) where they gave the asymptotically almost sure lower bounds and upper bonds on the size of maximum induced matchings in such graphs. The asymptotically almost sure lower bounds were achieved by analysing a degree-greedy algorithm using the differential equation method, whilst the asymptotically almost sure upper bounds were obtained by a direct expectation argument. In this paper, using the small subgraph conditioning method, we will show the asymptotically almost sure existence of an induced matching of certain size in random d-regular graphs, for d ∈ {3,4, 5}. This result improves the known asymptotically almost sure lower bound obtained by Duckworth et al.(2002).