In this paper, we deal with recursive constructions for incomplete group divisible designs (IGDDs). Denoting $GD[k,1,v;uv]–GD[k,1,n;un]$ by $(u,k)-IGD[v,n]$, we will prove, as an application, that a $(7,4)-IGD[v,n]$ exists if and only if $v\ge 3n$ and $\text{v – n}\equiv 0(\mathrm{mod}2)$.

We investigate the labellings of sum graphs, necessary conditions for a graph to be a sum graph, and the range of edge numbers of sum graphs.

A set $S$ of vertices of a graph is $k$-independent if each vertex in $S$ is adjacent to at most $k-1$ other vertices in $S$. A graph $G$ is well-$k$-covered if every maximal $k$-independent set is maximum. We shall characterize the well-$k$-covered trees and for $k=2$ all such graphs of girth $8$ or more.

We derive a first-order recurrence for $a_n(t)=\sum _{k=0}^n\frac{(-1{)}^{n-k}}{1+tk}(\genfrac{}{}{0ex}{}{n}{k})$ ($t$ fixed $t\ne -\frac{1}{m},m\in \mathbb{N}$). The first-order recurrence yields an alternative proof for Riordan’s theorem: $a_n(t)={(\genfrac{}{}{0ex}{}{1/t+n}{n})}^{-1}(-1{)}^n$ and also yields the ordinary generating function $\sum _{n=0}^{\mathrm{\infty }}{a}_n(t)x^n$ for $t\in \mathbb{N}.$From the latter, one easily computes $\sum _{n=0}^{\mathrm{\infty }}{a}_n(t)$ which turns out to be the well-known $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{n+1}=\ln 2$ for $t=1$. For $t=2$, we get $\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{2n}{(2n+1)}=\frac{\ln (\sqrt{2}+1)}{\sqrt{2}}$.

Avis has shown that the number of vertices of a minimal triangle-free $5$-chromatic graph is no fewer than $19$. Mycielski has shown that this number is no more than $23$. In this paper, we improve these bounds to $21$ and $22$, respectively.

By a refinement of a rank argument used to prove a directed version of the Graham-Pollak theorem, we show that $n$ bicliques are needed to partition the arc-set of the complement of a directed cycle.

In this paper, we obtain a polynomial inequality of degree three in $m$ (the number of constraints), with coefficients involving the parameters ${\mu }_i$’s, on the existence of balanced arrays of strength four and with two symbols. Applications of the inequality to specific balanced arrays for obtaining an upper bound on the number of constraints are also discussed.

Let $r(G)$ denote the rank, over the field of rational numbers, of the adjacency matrix of a graph $G$. Van Nuffelen and Ellingham have obtained several inequalities which relate $r(G)$ to other graph parameters such as chromatic number, clique number, Dilworth number, and domination number. We obtain additional results of this type. Our main theorem is that for graphs $G$ having no isolated vertices, $OIR(G)\le r(G)$, where $OIR(G)$ denotes the upper open irredundance number of $G$.

Let $D$ denote any balanced ternary design with block size three, index two, and ${\rho }_2=1$ (that is, with each element occurring repeated in just one block). This paper shows that there exists such a design $D$ on $V$ elements containing exactly $k$ pairs of repeated blocks if and only if $V\equiv 0(\mathrm{mod}3)$, $0\le k\le t_V=\frac{1}{6}V(V-3),k\ne t_V–1,\text{and}(k,V)\ne (1,6)$.

For each integer $v\ge 0$ and each $\lambda \in \{4,5,7,8\}$, the possible numbers of distinct blocks in a triple system of order $v$ and index $\lambda$ is determined. This essentially completes the determination of possible support sizes for triple systems with $\lambda \le 8$.

If $n$ is an integer, $n\ge 2$, and $u$ and $v$ are vertices of a graph $G$, then $u$ and $v$ are said to be $K_n$-adjacent vertices of $G$ if there is a subgraph of $G$, isomorphic to $K_n$, containing $u$ and $v$. A total $K_n$-dominating set of $G$ is a set $D$ of vertices such that every vertex of $G$ is $K_n$-adjacent to a vertex of $D$. The total $K_n$-domination number ${\gamma }_{K_n}^t(G)$ of $G$ is the minimum cardinality among the total $K_n$-dominating sets of vertices of $G$. It is shown that, for $n\in \{3,4,5\}$, if $G$ is a graph with no $K_n$-isolated vertex, then ${\gamma }_{K_n}^t(G)\le (2p)/n$. Further, $K_n$-connectivity is defined and it is shown that, for $n\in \{3,4\}$, if $G$ is a $K_n$-connected graph of order $\ge n+1$, then ${\gamma }_{K_n}^t(G)\le (2p)/(n+1)$. We establish that the upper bounds obtained are best possible.

Let $D$ and ${\overline{D}}^d$ be two designs such that there is a joint embedding $D^{\prime }$ and ${\overline{D}}^{\prime }$ of $D$ and $\overline{D}$ in a finite projective plane $\pi$ of order $n$ such that the points of $D^{\prime }$ and the lines of ${\overline{D}}^{\prime }$ are mutually all of the exterior elements of each other. We show that there is a tactical decomposition $T$ of $\pi$, two of the tactical configurations of which are $D^{\prime }$ and ${\overline{D}}^{\prime }$, and determine combinatorial restrictions on $n$ and the parameters of $D$ and ${\overline{D}}^d$. We also determine the entries of the incidence matrices of $T$.

The Josephus problem is concerned with anticipating which will be the last elements left in the ordered set $\{1,2,\dots ,n\}$ as successive $m$th elements (counting cyclically) are eliminated. We study the set of permutations of $\{1,2,\dots ,n\}$ which arise from the different orders of elimination as $m$ varies, and give a criterion based on the Chinese Remainder Theorem for deciding if a given permutation can be interpreted as arising as a given order of elimination for some step size $m$ in a Josephus problem.

In a secret sharing scheme, a dealer has a secret. The dealer gives each participant in the scheme a share of the secret. There is a set $\mathrm{\Gamma }$ of subsets of the participants with the property that any subset of participants that is in $\mathrm{\Gamma }$ can determine the secret. In a perfect secret sharing scheme, any subset of participants that is not in $\mathrm{\Gamma }$ cannot obtain any information about the secret. We will say that a perfect secret sharing scheme is ideal if all of the shares are from the same domain as the secret. Shamir and Blakley constructed ideal threshold schemes, and Benaloh has constructed other ideal secret sharing schemes. In this paper, we construct ideal secret sharing schemes for more general access structures which include the multilevel and compartmented access structures proposed by Simmons.

It is shown that the collection of all ${\displaystyle (\genfrac{}{}{0ex}{}{12}{5})}$ quintuples chosen from a set of twelve points can be partitioned into twelve mutually disjoint $4-(11,5,1)$ designs in precisely $24$ non-isomorphic ways. The results obtained are then used to show that the collection of all ${\displaystyle (\genfrac{}{}{0ex}{}{13}{6})}$ hextuples chosen from a set of thirteen points cannot be partitioned into thirteen mutually disjoint $5-(12,6,1)$ designs.

The set of Lyndon words of length $n$, ${\mathrm{\Lambda }}_n$, is the set obtained by choosing those strings of length $n$ over any finite alphabet $\mathrm{\Sigma }$ of cardinality $\sigma$ which are lexicographically least in the primitive or aperiodic equivalence classes determined by cyclic permutation. It is well-known that ${\mathrm{\Lambda }}_n$ is a maximal synchronizable code with bounded synchronization delay for fixed word length $n$. If the Lyndon words of length $n$ are represented as vertices of the $n$-cube, we show that they form a connected set for arbitrary alphabets. Indeed, we show that between any two Lyndon words, there is a path consisting of at most $2n$ Lyndon words in the $n$-cube. Further, we show that there always exists a path of $n(\sigma –1)–1$ Lyndon words in the $n$-cube.

The conjugation relation among the subspaces of a finite unitary geometry and its properties are studied. Then they are used to find some enumeration formulas for the subspaces of the unitary geometry, to prove a type of transitivity of the unitary group, to construct Partially Balanced Incomplete Block (PBIB) designs, and to establish the isomorphism between some known PBIB designs.

Incomplete group divisible designs (IGDDs) are the group divisible designs (GDDs) missing disjoint sub-GDDs, which need not exist. We denote by ${\text{IGDD}}_{\text{u}}^{\text{k}}(v,n)$ the design $\text{GDD}[k,1,v;uv]$ missing a sub-$\text{GDD}[k,1,n;un]$. In this paper, we give the necessary condition for the existence of ${\text{IGDD}}_{\text{u}}^{\text{k}}(v,n)$ and prove that the necessary condition is also sufficient for $k=3$.

The $vertexintegrity$ of a graph $I(G)$, is given by $I(G)=\underset{V^{\prime }}{min}(|V^{\prime }|+m(G–V^{\prime }))$ where $V^{\prime }\subseteq V(G)$ and $m(G–V^{\prime })$ is the maximum order of a component of $G–V^{\prime }$. The $edgeintegrity$, $I^{\prime }(G)$, is similarly defined to be $I^{\prime }(G)=\underset{E^{\prime }}{min}(|E^{\prime }|+m(G–E^{\prime }))$. Both of these are measures of the resistance of networks to disruption. It is shown that for each positive integer $k$, the family of finite graphs $G$ with $I^{\prime }(G)\le k$ is a lower ideal in the partial ordering of graphs by immersions. The obstruction sets for $k\le 4$ are determined and it is shown that the obstructions for arbitrary $k$ are computable. For every fixed positive integer $k$, it is decidable in time $O(n)$ for an arbitrary graph $G$ of order $n$ whether $I(G)$ is at most $k$, and also whether $I^{\prime }(G)$ is at most $k$. For variable $k$, the problem of determining whether $I^{\prime }(G)$ is at most $k$ is shown to be NP-complete, complementing a similar previous result concerning $I(G)$.

For positive integers $d$ and $m$, let $P_{d,m}(G)$ denote the property that between each pair of vertices of the graph $G$, there are $m$ openly disjoint paths of length at most $d$. A collection of such paths is called a $Mengerpathsystem$. Minimal conditions involving various combinations of the connectivity, minimal degree, sum of degrees, and unions of neighborhoods of pairs of nonadjacent vertices that insure the existence of Menger path systems are investigated. For example, if for fixed positive integers $d\ge 2$ and $m$, a graph $G$ has order $n$, connectivity $k\ge m$, and minimal degree $\delta >(n–(k–m+1)(d–2))/2+m–2$, then $G$ has property $P_{d,m}(G)$ for $n$. Also, if a graph $G$ of order $n$ satisfies $NC(G)>5n/(d+2)+2m$, then $P_{d,m}(G)$ is satisfied. (A graph $G$ satisfies $NC(G)\ge t$ if the union of the neighborhoods of each pair of nonadjacent vertices is at least $t$.) Other extremal results related to Menger path systems are considered.