SOME RESULTS ON SET COLORINGS OF DIRECTED TREES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. A set coloring of the digraph D is an assignment (function) of distinct subsets of a finite set X of colors to the vertices of the digraph, where the color of an arc, say (u,v) is obtained by applying the set difference from the set assigned to the vertex v to the set assigned to the vertex u which arc also distinct. A set coloring is called a strong set coloring if sets on the vertices and arcs are distinct and together form the set of all non empty subsets of X . A set coloring is called a proper set coloring if all the non empty subsets of X are obtained on the arcs of D. A digraph is called a strongly set colorable (properly set colorable) if it admits a strong set coloring (proper set coloring). In this paper we find some classes of directed trees which admit a strong set coloring and construction of strongly set colorable directed tree − → T ′ n .


INTRODUCTION
In this paper, we consider only finite simple digraphs. For all notations we follow Harary [1].
The notion of set coloring of a graph has been introduced by Hegde [2] in 2009. Further Hegde and Sumana [4] determined the set coloring number of certain graphs.
The concept of set colorings of graph was then extended to digraphs by Hegde and Castelino [3]. Definition 1.1. Given a digraph D = (V, E) with a non empty set X of n colors and m arcs, a function f : V → 2 X can be defined as the assignment of the colors f (v) to each of the vertices v ∈ V and given such a function f on the vertex set V ,we define f * : E → 2 X which assigns colors to the arcs e = uv ∈ E , f * (e) = f (v) − f (u).
A digraph D is said to be a set colorable if both f and f * are injective functions. A digraph D is said to be properly set colorable if it is set colorable with f * (E) = 2 X \ / 0 and D is strongly set They also determined the necessary condition for strong(proper) set colorings of digraphs.
Definition 1.2. Set coloring number [3],σ (D) of a digraph D is the least cardinality of a set X with respect to which D has a set coloring. Further, if f : V → 2 X is a set coloring of D with |X| = σ (D) we call f an optimal set coloring of D.
integer not less than the real x, and bounds are best possible.
In this paper, we find the set coloring number of unipath, some classes of digraphs which admit a strong(proper) set coloring and construction of a strongly set colorable directed tree.

SET COLORING NUMBER OF A DIGRAPH
In this section we find set coloring number of unipath.  Proof. Let the vertices of − → P 2 n be denoted by Let us assume that there exist a set coloring ( f , f * ) of − → P 2 n with respect to a set X of |X| = n and both f and f * are injective functions. That is sum of the number vertices and the number edges greater than 2 n which contradicts the fact that |X| = n. Therefore

STRONGLY (PROPERLY) SET COLORABLE DIRECTED TREES
In this section we present some results on strong(proper) set colorings of some classes of directed trees.
Cn is a directed tree obtained by joining each vertex of the unipath to a pendent vertex whose in degree is zero.
Proof. A directed centipede − → Cn has n vertices and n − 1 arcs. Let − → Cn be strongly set colorable directed tree with respect to a set X having k colors. Then|V ( .., k},the full set of X and X 2 is a subset containing k − 1 elements of X which doesn't contain the element a, a ∈ X. Then assign the set X 1 to the sink of − → Cn, that is vertex v of − → Cn, where od(v) = 0. Also assign the set X 2 to the vertex say u adjacent to v and id(u) = 0 and the remaining subsets of X to the n − 2 vertices of − → Cn. Then one can observe that the elements on the arcs are also subsets of X and together form the set of all nonempty subsets of X. Hence − → Cn is strongly set colorable.  Proof. A directed centipede − → Cn has n vertices and n − 1 arcs. Let − → Cn be properly set colorable directed tree with respect to a set X having k colors.
Strongly set coloring of a centipede.
Conversely, let w 0 be the root vertex of − → B (n, k) and w 1 , w 2 , w 3 , ..., w k be the central vertices of the k − stars joining the central vertex. Let u i,1 , u i,2 , u i,3 , ..., u i,n denote the pendent vertices joining w i , 1 ≤ i ≤ k. Let X be a non empty set with |X| = k. Let X 1 be the full set of X and X 2 be the (k − 1)-element set of X. Then we define a mapping f : Conversely, let − → Lb be a directed lobster such that n = 2 k−1 . Let v be the central vertex of − → Lb and od(v)=0. Let X 1 = {1, 2, ..., k}, the full set of X. Let P be the longest path from v.
Then assign (k − 1)-elements subsets of X say, A to the vertices of P. Let N 1 be the set of all vertices which are at a distance one from P. Then assign remaining (k − 1)-elements subsets of X together with (k − 2)-elements subsets of X other than A say, B to the vertices of N 1 . Let  Since f (