A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by χ=(G). It is interesting to note that if a graph G is equitably k-colorable, it does not imply that it is equitab...

The minimum number of total independent sets of V ∪ E of graph G(V,E) is called the total chromatic number of G, denoted by χ′′(G). If difference of cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V ∪E is called the equitable total chromatic number, and denoted by χ′′ =(G). In this paper we consider equitable total c...

There is a rich literature on conditions guaranteeing that certain sums of residue classes cover certain other residue classes modulo some integer N . Often it is of particular importance for applications to know that the class 0 mod N can be represented as a sum of the studied residue classes, and the zero class is often the most difficult case. A well-known result along these lines is the fam...

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ=(G). In this paper the problem of determinig the value of equitable chromatic number for multic...

McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that B ≤s A if sA − sB is Schurpositive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length a or (a+ 1) for a ≥ 2. In this paper we confirm the conjecture of McNamara and Py...

background and purpose : there has been no report in which preoperative evaluation of adc for prediction of post therapeutic outcome was presented. the aim of our study was to retrospectively evaluation the use of the apparent diffusion coefficient (adc) for the prediction of malignancy and the outcome of malignant astrocytic tumors. methods: we reviewed mri of 26 male and 14 female patients wi...