An algorithm is presented in which a polynomial deck, \(\mathcal{P}D\), consisting of \(m\) polynomials of degree \(m-1\), is analysed to check whether it is the deck of characteristic polynomials of the one-vertex-deleted subgraphs of the line graph, \(H\), of a triangle-free graph, \(G\). We show that if two necessary conditions on \(\mathcal{P}D\), identified by counting the edges and triangles in \(H\), are satisfied, then one can construct potential triangle-free root graphs, \(G\), and by comparing the polynomial decks of the line graph of each with \(\mathcal{P}D\), identify the root graph.

Let \(\sigma_2(G) = \min\{d_G(u)+d_G(v) | u,v \in V(G), u,v \notin E(G)\}\) for a non-complete graph \(G\). An \([a, b]\)-factor of \(G\) is a spanning subgraph \(F\) with minimum degree \(\delta(F) \geq a\) and maximum degree \(\Delta(F) \leq b\).
In this note, we give a partially positive answer to a conjecture of M. Kano. We prove the following results:

Let \(G\) be a 2-edge-connected graph of order \(n\) and let \(k \geq 2\) be an integer. If \(\sigma_2(G) \geq {4n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor if \(k\) is even and a 2-edge-connected \([2, k + 1]\)-factor if \(k\) is odd.
Indeed, if \(k\) is odd, there exists a graph \(G\) which satisfies the same hypotheses and has no 2-edge-connected \([2, k]\)-factor.
Nevertheless, we have shown that if \(G\) is 2-connected with minimum degree \(\delta(G) \geq {2n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor.

The Ramsey numbers \(r(C_4,G)\) are determined for all graphs \(G\) of order six.

In a \(t-(v,k,\lambda)\) directed design, the blocks are ordered \(k\)-tuples and every ordered \(t\)-tuple of distinct points occurs in exactly \(\lambda\) blocks (as a subsequence). We show that a simple \(3-(v,4,2)\) directed design exists for all \(v\). This completes the proof that the necessary condition \(\lambda v\equiv 0 \pmod 2\) for the existence of a \(3-(v,4,\lambda)\) directed design is sufficient.

We give a conjecture for the total chromatic number \(\chi_T\) of all Steiner systems and show its relationship to the celebrated Erdős, Faber, Lovász conjecture. We show that our conjecture holds for projective planes, resolvable Steiner systems and cyclic Steiner systems by determining their total chromatic number.

We propose a number of problems about \(r\)-factorizations of complete graphs. By a completely novel method, we show that \(K_{2n+1}\) has a \(2\)-factorization in which all \(2\)-factors are non-isomorphic. We also consider \(r\)-factorizations of \(K_{rn+1}\) where \(r \geq 3\); we show that \(K_{rn+1}\) has an \(r\)-factorization in which the \(r\)-factors are all \(r\)-connected and the number of isomorphism classes in which the \(r\)-factors lie is either \(2\) or \(3\).

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.