A Perfect star packing in a fullerene graph G is a spanning subgraph of G whose every component is isomorphic to the star graph K_1,3. A Perfect pseudo matching of a fullerene graph G is a spanning subgraph H of G such that each component of H is either K_2 or K_1,3. In this paper, we examine the number of Perfect star packing in (3,6)-fullerene graphs and Perfect pseudo matching in chamfered fullerene graphs.

‎A Perfect star packing in a given graph $G$ can be defined as a spanning subgraph of $G$‎, ‎wherein each component is isomorphic to the star graph $K_{1,3}$‎. ‎A Perfect star packing of a fullerene graph $G$ is of type $P0$ if all the centers of stars lie on hexagons of $G$‎. ‎Many fullerene graphs arise from smaller fullerene graphs by applying some transformations‎. ‎In this paper‎, ‎we introduce two transformations for fullerene graphs that have the Perfect star packing of type $P0$ and examine some characteristics of the graphs obtained from this transformation‎.

A tree T of order n is called k-placeable if there are k edgedisjoint copies of T into Kn. In this paper we prove some results about 4-placement of rooted trees.

The size of database and minimum number of visible stars in the field of view of star sensor are two important, influential and contradictory parameters that should be considered in design of star sensor. In this regard, the purpose of this paper is to unify the database using the uniform distribution of points on the celestial sphere with the triangulation method. For this purpose, the choice of the suitable star catalog, minimum suitable magnitude and elimination of double stars are the other steps of the uniformity process that is carried out in this study. Thus, the results of the investigations showed that Delaunay's triangulation method is faster and more accurate than the geodesic grid. Also, by simulating and performing Monte Carlo tests to count the number of stars observed in the different FOVs of a typical sensor, it was found that Delaunay's triangulation leads to a significant reduction of the probability of viewing the high density of the catalog stars in the field of view, so that the probability of observation more than 25 stars in all possible FOVs has reached to zero. On the other hand, for observing 4 or more than 4 stars at a confidence level more than 95% in non-uniform catalog, the field of view needs to be at least 12. 5 degrees, while in uniform database; this field is slightly increased to more than 13 degrees. In other words, uniformity has increased the minimum field of view needed to see the minimum number of required stars.

In a graphG= (V; E), a subset S Í V (G) is said to be an open packing setof G if no two vertices of S have a common neighbour inG: The maximum cardinality of an open packing set is called the open packing numberof G and is denoted by ro (G). This parameter has been studied in [5], [6], [7] and [8]. In this paper, we characterize the graphs G with ro (G) =n - 2, ro (G) =n -w (G) and ro (G) =n - D (G), where n, w (G) and D (G) denote the order, clique number and the maximum degree of G. Also, we discuss the open packing number for split graphs.