Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or G contains a G2. A complete bipartite graph K1,n is called a star. The kipas ̂ Kn is the graph obtained from a path of order n by adding a new vertex and joining it to all the vertices of the path. Alternatively, a kipas is a wheel with one edg...

A proper vertex coloring of a graphG is called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) ∈ L(v). If G is L-star-colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is called k-star-choosable. The star list chromatic number of G, ...

We show that for n ~ 2, every n-vertex tree has a star forest of order exceeding n /2. There are many theorems in graph theory in which a graph is required to have a particular subgraph. A stronger requirement, however, is that it be an induced subgraph: i.e. a subgraph in which two vertices must be adjacent if they are adjacent in the big graph. A common problem of Ramsey theory, for instance,...

The class of star graphs is a popular topology for interconnection networks. However it has certain deficiencies. A class of generalization of star graphs called (n, k)-star graphs was introduced in [12] to address these issues. In this paper we will consider the vertex-connectivity of the directed (n, k)-star graph, −−→ Sn,k, given in [8], and show that it is maximally connected.

The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kr such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-critical Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

Motivated by applications that require mechanisms for describing the structure of object-oriented programs, adaptive star grammars are introduced, and their fundamental properties are studied. In adaptive star grammars, rules are actually schemata which, via the cloning of so-called multiple nodes, may adapt to potentially infinitely many contexts when they are applied. This mechanism makes ada...

A p. q graph G = V,E is said to be a square graceful graph ifthere exists an injective function f: V G → 0,1,2,3,... , q such that the induced mapping fp : E G → 1,4,9,... , q 2 defined by fp uv = f u − f v is an injection. The function f is called a square graceful labeling of G. In this paper the square graceful labeling of the caterpillar S X1,X2 ,... ,Xn , the graphs Pn−1 1,2,...n ,mK1,n ∪ ...

For a subset W of vertices of an undirected graph G, let S(W ) be the subgraph consisting of W , all edges incident to at least one vertex in W , and all vertices adjacent to at least one vertex in W . If S(W ) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that fo...

The arrangement graph A n , k , which is a generalization of the star graph (n k = l), presents more flexibility than the star graph in adjusting the major design parameters: number o f nodes, degree , and diameter. Previously the arrangement graph has proven hamiltonian. In this p a p e r 'we further show that the arrangement graph remains hamiltonian even if it is faulty. Let IFe/ and IF,[ de...