A decomposition of optimal linear block codes with minimum distance \(d = 4\) and length \(4L\) into two subcodes is given such that one of the subcodes is an optimal length \(L\) code with minimum Hamming distance \(4\) and the other is a quasi-cyclic code of index \(4\). It is shown that the \(L\)-section minimal trellis diagram of the code is the product of the minimal trellis diagrams of the subcodes.

We consider the rank of the adjacency matrix of some classes of regular graphs that are transformed under certain unary operations. In particular, we study the ranks of the subdivision graph, the connected cycle graph, the connected subdivision graph, and the total graph of the following families of graphs: cycles, complete graphs, complete bipartite and multipartite graphs, circulant graphs of degrees three and four, and some Cartesian graph products.

It is proved that the total chromatic number of any series-parallel graphs of degree at least \(3\) is \(\Delta(G)+1\).

We show that, in any coloring of the edges of \(K_{36}\), with two colors, there exists a triangle in the first color or a monochromatic \(K_{10}-e\) (\(K_{10}\) with one edge removed) in the second color, and hence we obtain a bound on the corresponding Ramsey number, \(R(K_3, K_{10}-e) \leq 38\). The new lower bound of \(37\) for this number is established by a coloring of \(K_{36}\) avoiding triangles in the first color and \(K_{10}-e\) in the second color. This improves by one the best previously known lower and upper bounds. We also give the bounds for the next Ramsey number of this type, \(42 \leq R(K_3, K_{11}-e) \leq 47\).

A subset \(S\) of \(V(G)\) is called a dominating set if every vertex in \(V(G) – S\) is adjacent to some vertex in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality taken over all dominating sets of \(G\). A dominating set \(S\) is called a tree dominating set if the induced subgraph \(\langle S\rangle\) is a tree. The tree domination number \(\gamma_{tr}(G)\) of \(G\) is the minimum cardinality taken over all minimal tree dominating sets of \(G\). In this paper, some exact values of tree domination number and some properties of tree domination are presented in Section [2]. Best possible bounds for the tree domination number, and graphs achieving these bounds are given in Section [3]. Relationships between the tree domination number and other domination invariants are explored in Section [4], and some open problems are given in Section [5].

If \(G\) is a tricyclic Hamiltonian graph of order \(n\) with maximum degree \(3\), then \(G\) has one of two forms, \(X(q,r,s,t)\) and \(Y(q,r,s,t)\), where \(q+r+s+t=n\). We find the graph \(G\) with maximal index by first identifying the graphs of each form having maximal index.

Let \(G = (V_1, V_2; E)\) be a bipartite graph with \(|V_1| = |V_2| = n \geq 2k\), where \(k\) is a positive integer. Let \(\sigma'(G) = \min\{d(u)+d(v): u\in V_1, v\in V_2, uv \not\in E(G)\}\). Suppose \(\sigma'(G) \geq 2k + 2\). In this paper, we will show that if \(n > 2k\), then \(G\) contains \(k\) independent cycles. If \(n = 2k\), then it contains \(k-1\) independent \(4\)-cycles and a \(4\)-path such that the path is independent of all the \(k-1\) \(4\)-cycles.

New results on the enumeration of noncrossing partitions with \(m\) fixed points are presented, using an enumeration polynomial \(P_m(x_1, x_2, \ldots, x_m)\). The double sequence of the coefficients \(a_{m,k}\) of each \(x^k_i\) in \(P_m\) is endowed with some important structural properties, which are used in order to determine the coefficient of each \(x^k_ix^l_j\) in \(P_m\).

This paper concerns a labeling problem of the plane graphs \(P_{a,b}\). We discuss the magic labeling of type \((1,1,1)\) and consecutive labeling of type \((1,1,1)\) of the graphs \(P_{a,b}\).

In this note, we prove that the largest non-contractible to \(K^p\) graph of order \(n\) with \(\lceil \frac{2n+3}{3} \rceil \leq p \leq n\) is the Turán’s graph \(T_{2p-n-1}(n)\). Furthermore, a new upper bound for this problem is determined.