Gronau, Mullin and Pietsch determined the exact closure of index one of all subsets \(K\) of \(\{3,\ldots,10\}\) 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 \(\Delta = 3\) and \(n_2 \geq 2\), 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 \(\Delta = 3\) 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 \(\gamma_t(G) \leq 4n/7\) 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, \(\delta(G) \geq 2\), and \(\gamma(G) \geq 4n/7\) is obtained. We establish that if \(G\) is a connected graph of size \(q\) with minimum degree at least 2, then \(\gamma_t(G) \leq (q + 2)/2\). 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 \(m \geq 3k – 3\), or \(m\) is even and \(2k – 2 \leq m \leq 3k – 4\). 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 \(2k\) (\(k \neq 3,4\)) is shown to be unique. A generalisation of our constructions to trades with blocks based on arbitrary simple graphs is also presented.