Let \(n \geq 1\) be an integer. The closed \(n\)-neighborhood \(N_n[u]\) of a vertex \(u\) in a graph \(G = (V, E)\) is the set of vertices \(\{v | d(u,v) \leq n\}\). The closed \(n\)-neighborhood of a set \(X\) of vertices, denoted by \(N_n[X]\), is the union of the closed \(n\)-neighborhoods \(N_n[v]\) of vertices \(u \in X\). For \(X \subseteq V(G)\), if \(N_n[x] – N_n[X – \{u\}] = \emptyset\), then \(u\) is said to be \(n\)-redundant in \(X\). A set \(X\) containing no \(n\)-redundant vertex is called \(n\)-irredundant. The \(n\)-irredundance number of \(G\), denoted by \(ir_n(G)\), is the minimum cardinality taken over all maximal \(n\)-irredundant sets of vertices of \(G\). The upper \(n\)-irredundance number of \(G\), denoted by \(IR_n(G)\), is the maximum cardinality taken over all maximal \(n\)-irredundant sets of vertices of \(G\). In this paper we show that the decision problem corresponding to the computation of \(ir_n(G)\) for bipartite graphs \(G\) is NP-complete. We then prove that this also holds for augmented split graphs. These results extend those of Hedetniemi, Laskar, and Pfaff (see [7]) and Laskar and Pfaff (see [8]) for the case \(n = 1\). Lastly, applying the general method described by Bern, Lawler, and Wong (see [1]), we present linear algorithms to compute the \(2\)-irredundance and upper \(2\)-irredundance numbers for trees.

Some properties of finite projective planes are used to obtain some new pairwise balanced designs with consecutive block sizes, by deleting configurations spanned by lines.

We give a short survey of the best known lower bounds on \(K(n, 1)\), the minimum cardinality of a binary code of length \(n\) and covering radius \(1\). Then we prove new lower bounds on \(K(n, 1)\), e.g.
\[K(n,1)\geq \frac{(5n^2-13n+66)2^n}{(5n^2-13n+46)(n+1)}\] when \(n \equiv 5 \pmod{6}\)
which lead to several numerical improvements.

In this paper, we study path-factors and path coverings of a claw-free graph and those of its closure. For a claw-free graph \(G\) and its closure \( cl(G)\), we prove:(1) \(G\) has a path-factor with \(r\) components if and only if \( cl(G)\) has a path-factor with \(r\) components,(2) \(V(G)\) is covered by \(k\) paths in \(G\) if and only if \(V( cl(G))\) is covered by \(k\) paths in \( cl(G)\).

Let \(G = (V,E)\) be a connected graph. Let \(\gamma_c(G), d_c(G)\) denote the connected domination number, connected domatic number of \(G\), respectively. We prove that \(\gamma_c(G) \leq 3d_c(G^c)\) if the complement of \(G\) is also connected. This confirms a conjecture of Hedetniemi and Laskar (1984), and Sun (1992). Examples are given to show that equality may occur.

A method of construction of quasi-multiple balanced incomplete block \((BIB)\) designs from certain group divisible designs is described. This leads to a series of quasi-multiple designs of symmetric BIB designs and new non-isomorphic solutions of designs listed as unknown in the tables of Mathon and Rosa \([{3,4}]\). In the process a series of semi-regular group divisible designs is also obtained.

In this paper, we construct two series of balanced incomplete block (BIB) designs with parameters:
\[v = \binom{2m-3}{2} ,r= \frac{(2m-5)!}{(m-1)!}, k= {m}\]
\[b=\frac{(2m-3)!}{2m(m-1)!} , \lambda = \frac{(2m-6)!}{(m-3)!}\]
and
\[v = \binom{2m+1}{2} , b = b_1(m+1), r = 2m(\overline{\lambda}_1-\overline{\lambda}_2), k = m^2\]
\[\lambda = (m-1)(\overline{\lambda}_1-2\overline{\lambda}_2+\overline{\lambda}_3)+m(\overline{\lambda}_2-\overline{\lambda}_3)\]
where \(k_1, b_1, \overline{\lambda}_i\) are parameters of a special \(4-(v, k, \lambda)\) design.

The strong chromatic index of a graph \(G\), denoted \(sq(G)\), is the minimum number of parts needed to partition the edges of \(G\) into induced matchings. The subset graph \(B_m(k)\) is the bipartite graph whose vertices represent the elements and the \(k\)-subsets of an \(m\) element ground set where two vertices are adjacent if and only if the vertices are distinct and the element corresponding to one vertex is contained in the subset corresponding to the other. We show that \(sq(B_m(k)) =\binom{m}{k-1}\) and that this satisfies the strong chromatic index conjecture by Brualdi and Quinn \([3]\) for bipartite graphs.

For a graph \(G\), if \(F\) is a nonempty subset of the edge set \(E(G)\), then the subgraph of \(G\) whose vertex set is the set of ends of edges in \(F\) is denoted by \(_G\). Let \(E(G) = \cup_{i \in I} E_i\) be a partition of \(E(G)\), let \(D_i = _G\) for each \(i\), and let \(\phi = (D_i | i \in I)\), then \(\phi\) is called a partition of \(G\) and \(E_i\) (or \(D_i\)) is an element of \(\phi\). Given a partition \(\phi = (D_i | i \in I)\) of \(G\), \(\phi\) is an admissible partition of \(G\) if for any vertex \(v \in V(G)\), there is a unique element \(D_i\) that contains vertex \(v\) as an inner point. For two distinct vertices \(u\) and \(v\), a \(u-v\) walk of \(G\) is a finite, alternating sequence \(u = u_0, e_1, u_1, e_2, \ldots, v_{n.1},e_n,u_n = v\) of vertices and edges, beginning with vertex \(u\) and ending with vertex \(v\), such that \(e_i = u_{i-1}u_i\) for \(i = 1, 2, \ldots, n\). A \(u-v\) string is a \(u-v\) walk such that no vertex is repeated except possibly \(u\) and \(v\), i.e., \(u\) and \(v\) are allowed to appear at most two times. Given an admissible partition \(\phi\), \(\phi\) is a string decomposition or \(SD\) of \(G\) if every element of \(\phi\) is a string.

In this paper, we prove that a \(2\)-connected graph \(G\) has an \(SD\) if and only if \(G\) is not a cycle. We also give a characterization of the graphs with cut vertices such that each graph has an \(SD\).

The cyclic chromatic number is the smallest number of colours needed to colour the nodes of a tournament so that no cyclic triple is monochromatic. Bagga, Beineke, and Harary \({[1]}\) conjectured that every tournament score vector belongs to a tournament with cyclic chromatic number \(1\) or \(2\). In this paper, we prove this conjecture and derive some other results.