Gronau, Mullin and Pietsch determined the exact closure of index one of all subsets $K$ of $\{3,\dots ,10\}$ which include $3$. We extend their results to obtain the exact closure of such $K$ for all indices.

For every connected, even-degree graph $G$ with $10$ or fewer edges, the problem of finding necessary and sufficient conditions for the existence of a decomposition of $K_v$ into edge-disjoint copies of $G$ is completely settled.

The Huffman coding scheme is a character-based algorithm in which every leaf node represents a character only. In this paper, we study three variations of the Huffman coding scheme for compressing 16-bit Chinese language. Although it is observed that IDC can generate the shortest code length among the three variations, but its empirical compression ratio is below 1.8, which is unsatisfactory. In order to achieve higher compression performance, i.e., compression ratio over 2, word-based compression algorithms should be employed. A possible way to develop word-based algorithms is to use the technique of cascading. Two kinds of algorithms are chosen for cascading. They are LZ algorithms and the Huffman coding scheme. LZ algorithms are used for finding repeating phrases while the Huffman coding scheme is used for encoding the phrases instead of characters. The experimental results show that the cascading algorithm of LZSSPDC outperforms a famous UNIX cascading compressor GZIP by 5\% on average.

Grotzsch conjectured that if $G$ is planar, bridgeless with $\mathrm{\Delta }=3$ and $n_2\ge 2$, then $G$ is of Class one. We prove that when $n_2=2$ the conjecture is equivalent to the statement: $G$ is $3$-critical if $G$ is planar, bridgeless with $\mathrm{\Delta }=3$ and $n_2=1$. Then we prove that the conjecture implies the Four Color Theorem.

In this paper, we prove that a $V(3,t)$ exists for any prime power $3t+1$, except when $t=5$, as no $V(3,5)$ exists.

In this paper, we survey some recent bounds on domination parameters. A characterisation of connected graphs with minimum degree at least 2 and domination number exceeding a third their size is obtained. Upper bounds on the total domination number, ${\gamma }_t(G)$, of a graph $G$ in terms of its order and size are established. If $G$ is a connected graph of order $n$ with minimum degree at least 2, then either ${\gamma }_t(G)\le 4n/7$ or $G\in \{C_3,C_5,C_6,C_{10}\}$. A characterisation of those graphs of order $n$ which are edge-minimal with respect to satisfying $G$ connected, $\delta (G)\ge 2$, and $\gamma (G)\ge 4n/7$ is obtained. We establish that if $G\$isaconnectedgraphofsize\text{(}q$ with minimum degree at least 2, then ${\gamma }_t(G)\le (q+2)/2$. Connected graphs $G$ of size $q$ with minimum degree at least 2 satisfying ${\gamma }_t(G)>q/2$ are characterised. Upper bounds on other domination parameters, including the strong domination number and the restrained domination number are presented. We provide a constructive characterisation of those trees with equal domination and restrained domination numbers. A constructive characterisation of those trees with equal domination and weak domination numbers is also obtained.

Necessary and sufficient conditions for the existence of a decomposition of $\lambda K_v$ into edge-disjoint copies of the Petersen graph are proved.

A $(v,k,t)$ trade $T=T_1–T_2$ of volume $m$ consists of two disjoint collections $T_1$ and $T_2$, each containing $m$ blocks ($k$-subsets) such that every $t$-subset is contained in the same number of blocks in $T_1$ and $T_2$. If each $t$-subset occurs at most once in $T_1$, then $T$ is called a Steiner $(k,t)$ trade. In this paper, the spectrum (that is, the set of allowable volumes) of Steiner trades is discussed, with particular reference to the case $t=2$. It is shown that the volume of a Steiner $(k,2)$ trade is at least $2k–2$ and cannot equal $2k–1$. We show how to construct a Steiner $(k,2)$ trade of volume $m$ when $m\ge 3k–3$, or $m$ is even and $2k–2\le m\le 3k–4$. For $k=5$ or $6$, the non-existence of Steiner $(k,2)$ trades of volume $2k+1$ is demonstrated, and for $k=7$, we exhibit a Steiner $(k,2)$ trade of volume $2k+1$. In addition, the structure of Steiner $(k,2)$ trades of volumes $2k–2$ and $2k$ ($k\ne 3,4$) is shown to be unique. A generalisation of our constructions to trades with blocks based on arbitrary simple graphs is also presented.

This paper characterizes a particular scheme of partially filled Latin squares and when they can be completed to full Latin squares. In particular, given an $n\times n$ array with the first $s$ rows and the first $d$ cells of row $s+1$ filled with $n$ distinct symbols in such a way that no symbol occurs more than once in any row or column, necessary and sufficient conditions are found for when this array can be completed to a full Latin square.

We give counterexamples for two theorems given for the integrity of prisms and ladders in [2] (Theorem 2.17 and Theorem 2.18 in [1]). We also compute the integrity of several special graphs.

We apply a lattice point counting method due to Blass and Sagan [2] to compute the characteristic polynomials for the subspace arrangements interpolated between the Coxeter hyperplane arrangements. Our proofs provide combinatorial interpretations for the characteristic polynomials of such subspace arrangements. In the process of doing this, we explore some interesting properties of these polynomials.

A graph of a puzzle is obtained by associating each possible position with a vertex and by inserting edges between vertices if and only if the corresponding positions can be obtained from each other in one move. Computational methods for finding the vertices at maximum distance $\delta$ from a vertex associated with a goal position are presented. Solutions are given for small sliding block puzzles, and methods for obtaining upper and lower bounds on $\delta$ for large puzzles are considered. Old results are surveyed, and a new upper bound for the 24-puzzle is obtained: $\delta \le 210$.

The total domination number ${\gamma }_t(G)$ of graph $G=(V,E)$ is the cardinality of a smallest subset $S$ of $V$ such that every vertex of $V$ has a neighbor in $S$. It is known that, if $G$ is a connected graph with $n$ vertices, ${\gamma }_t(G)\le \lfloor 2n/3\rfloor$. Graphs achieving this bound are characterized.

In this paper we define the imbalance of equi-replicate incomplete block designs. We prove that the imbalance measure of an equi-replicate incomplete block design has a lower bound, and this bound is attained if and only if the design is a 2-concurrence design. This result allows one to formulate the construction of 2-concurrence designs as an optimization problem.

Until quite recently, very few weakly completable critical sets were known. The purpose of this note is to prove the existence of at least one Latin square of each order greater than four in which a weakly completable set exists. This is done by actual construction of such a square. Non-existence of weakly completable sets in Latin squares of orders 2, 3, and 4 is already known.

In our recent paper Necessary and sufficient conditions for some two variable orthogonal designs in order 44, Koukouvinos, Mitrouli and Seberry leave 7 cases unresolved. Using a new algorithm given in our paper A new algorithm for computer searches for orthogonal designs by the present four authors we are able to finally resolve all these cases.

This note records that the necessary conditions for the existence of two variable designs constructed using four circulant matrices are sufficient. In particular, of 484 potential cases, 404 cases have been found, 68 cases do not exist, and 12 cases cannot be constructed using four circulant matrices.

We determine the number of non-isomorphic triple systems with bipoints in those cases for which the total number of triples does not exceed 20.

Temporal load-balancing – “spreading out” the executions of tasks over time — is desirable in many applications. A form of temporal load-balancing is introduced: scheduling to maximize minimum inter-completion time (MICT-scheduling). It is shown that MICT-scheduling is, in general, NP-hard. A number of restricted classes of task systems are identified, which can be efficiently MICT-scheduled.

Given a finite-dimensional vector space $V$ over a finite field $F$ of odd characteristic, and equipping $V$ with an orthogonal (symplectic, unitary) geometry, the following two questions are considered:

\begin{enumerate}
\item Given some linearly independent vectors $w_1,w_2,\dots ,w_k\in V$ and the $k\times k$ matrix $A=(⟨w_i,w_j⟩)$, and given scalars ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k,\beta \in F$, how many vectors $v\in V$, not in the linear span of $w_1,w_2,\dots ,w_k$, satisfy $⟨w_i,v⟩={\alpha }_i\$(\text{(}i=1,2,\dots ,k\$)and\text{(}⟨v,v⟩=\beta$?

\item Given a $k\times k$ matrix $A=({\lambda }_{ij})$ with entries from $F\$,howmany\text{(}k$-tuples $(v_1,v_2,\dots ,v_k)$ of linearly independent vectors from $V$ satisfy $⟨v_i,v_j⟩={\lambda }_{ij}$ ($i,j=1,2,\dots k$)?
\end{enumerate}

An exact answer to the first question is derived. Here there are two cases to consider, depending on whether or not the column vector $({\alpha }_i)$ is in the column space of $A$. This result can then be applied iteratively to address the second question.

Let $G$ be a finite graph and let $\mu$ be an eigenvalue of $G$ of multiplicity $k$. A star set for $\mu$ may be characterized as a set $X$ of $k$ vertices of $G$ such that $\mu$ is not an eigenvalue of $G–X$. It is shown that if $G$ is regular then $G$ is determined by $\mu$ and $G–X$ in some cases. The results include characterizations of the Clebsch graph and the Higman-Sims graph.