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A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $ n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$.

In this paper, we show that for all $v\equiv 0,1$ (mod 5) and $v\geq 15$, there exists a super-simple $(v,5,2)$ directed design, also for these parameters there exists a super-simple $(v,5,2)$ directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes., Comment: arXiv admin note: substantial text overlap with arXiv:1508.00098

The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}., Comment: 14 pages

A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_{i}$ (for $1 \leq i \leq \mu$). A $\mu$-way $(v, k, t)$ trade is called $\mu$-way $(v, k, t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_{1}$ $(T_{j},\ j \geq 2)$. A $\mu$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_{1}$ $(T_{j},~ j \geq 2)$. In this paper we characterize the $3$-way $3$-homogeneous $(v,3,2)$ Steiner trades of volume $v$. Also we show how to construct a $3$-way $d$-homogeneous $(v,3,2)$ Steiner trade for $d\in \{4,5,6\}$, except for seven small values of $v$., Comment: 11 pages. Utilitas Math.(to appear)

In this paper, we show that for all v\pmod 1 (mod 3), there exists a super- simple (v, 4, 2) directed design. Also, we show that for these parameters there exists a super-simple (v, 4, 2) directed design whose each defining set has at least a half of the blocks., Comment: 17 pages. Discrete Applied Mathematics(accepted)