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Graph theory can be said as one of the branches of applied mathematics that is currently developing. A problem can be explained in detail by simplifying it with graph theory. This paper concerns with graphs that are used in everyday life as a network system to describe parts of a structure in order to make it simpler and easier to understand. The network flow system is then applied in many disciplines, including applied mathematics, computer science, operations research, management, and engineering. [ABSTRACT FROM AUTHOR]

The problem of finding of the partition dimension for a general graph is an NP-hard problem. Therefore, many studies of partition dimension of graphs are focused on particular classes of graphs. In this paper, we are interested in determining the partition dimension for the resulting graphs produced by the corona product between two graphs. In specific, we consider the partition dimension of the corona graphs K1⊙mKn, Pm⊙K1,n and K1,m⊙Kn where Pm,K1,n and Kn are a path order m, a star order n+1 and a complete graph order n, respectively. [ABSTRACT FROM AUTHOR]

For a graph G(V,E), an edge-covering of G is a family of different subgraphs H1,...Hk such that any edge of E belongs to at least one of the subgraphs Hi, 1 ≤ i ≤ k. If every Hi is isomorphic to a given graph H, then G admits an H-covering. Graph G is said to be H-magic if G has an H-covering and there is a total labeling f:V∪E→{1,2,...|V|+|E|} such that for each subgraph H′ = (V′,E′) of G isomorphic to H,ΣνƸV′f(ν)+ΣeƸE′f(e) is fixed constant. Furthermore, if f(V) = }1,2,...|V|{ then G is called H-supermagic. The sum of all vertex and edge labels on H, under a labeling f, is denoted by Σf(H). In this paper we study H-supermagic labeling for some classes of trees such as a double star, a caterpillar, a firecracker and a banana tree. [ABSTRACT FROM AUTHOR]

For any given two graphs G and H, we use the notation F→(G,H) to mean that in any red-blue coloring of the edges of F, the following must hold: F contains either a red subgraph G or a blue subgraph H. A graph F is a Ramsey (G,H)-minimal graph if F→(G,H) but F*↛(G,H) for any proper subgraph F* of F. Let R(G,H) be the class of all Ramsey (G,H)-minimal graphs. In this paper, we derive the properties of graphs belonging to R(3K2,P3). By using these properties we determine all graphs belonging to this class. [ABSTRACT FROM AUTHOR]

An f-coloring of graph G(V,E) is a generalized edge-coloring such that every vertex ν in V has at most f(ν) edges colored with a same color. There are many applications of the f-coloring, for instance we can use it in order to solve some scheduling problems and some network design problems. The minimum number of colors needed to f-color G is called an the f-chromatic index of G, denoted by χ′f(G). Any graph G has f-chromatic index equal to Δf(G) or Δf(G)+1, where Δf(G) = maxνƸV{[ceiling_left]d(ν)/f(ν)[ceiling_right]}. G is called in the class-1, denoted by G Ƹ Cf 1, if χ′f(G) = Δf(G); otherwise G is called in the class-2, denoted by G Ƹ Cf 2. In this paper, we show that the corona product of Cn with Sm is in Cf 1. Besides that, we characterize the corona product of Cn with either Wn or Kn based on f-coloring. [ABSTRACT FROM AUTHOR]

For given two graphs G1 and G2, the size Ramsey multipartite numbers mj(G1,G2) is the smallest integer t such that every factorization of graph Kj×t := F1 + F2 satisfies the following condition: either F1 contains G1 as a subgraph or F2 contains G2 as a subgraph. In this paper, we derive some lower bound for the size multipartite Ramsey numbers mj(Pn,Kj×b) with integers j,n ≥ 3 and b ≥ 2. [ABSTRACT FROM AUTHOR]