In this paper, we investigate the basis number for the wreath product of wheels with paths. Also, as a related problem, we construct a minimum cycle basis of the same.

Let\(ex(m, C_{\leq n})\) denote the maximum size of a graph of order \(m\) and girth at least \(n+1\), and \(EX(m, C_{\leq n})\) be the set of all graphs of girth at least \(n+1\) and size \(ex(m, C_{\leq n})\). The Ramsey number \(R(C_{\leq n}, K_m)\) is the smallest \(k\) such that every graph of order \(k\) contains either a cycle of order \(n\) for some \(3 \leq l \leq n\) or a set of \(m\) independent vertices. It is known that \(ex(2n, C_{\leq n}) = 2n + 2\) for \(n \geq 4\), and the exact values of \(R(C_{\leq n}, K_m)\) for \(n \geq m\) are known. In this paper, we characterize all graphs in \(EX(2n, C_{\leq n})\) for \(n \geq 5\), and then obtain the exact values of \(R(C_{\leq n}, K_m)\) for \(m \in \{n, n+1\}\).

Since their desirable features, variable-weight optical orthogonal codes (VWOOCs) have found wide ranges of applications in various optical networks and systems. In recent years, optimal \(2\)-CP\((W, 1, Q; n)\)s are used to construct optimal VWOOCs. So far, some works have been done on optimal \(2\)-CP\((W, 1, Q; n)\)s with \(w_{\max} \leq 6\), where \(w_{\max} = \max\{w: w \in W\}\). As far as the authors are aware, little is known for explicit constructions of optimal \(2\)-CP\((W, 1, Q; n)\)s with \(w_{\max} \geq 7\) and \(|W| = 3\). In this paper, two explicit constructions of \(2\)-CP\((\{3, 4, 7\}, 1, Q; n)\)s are given, and two new infinite classes of optimal VWOOCs are obtained.

In this study, it has been researched which Euclidean regular polyhedrons are also taxicab regular and which are not. The existence of non-Euclidean taxicab regular polyhedrons in the taxicab \(3\)-space has also been investigated.

As a generalization of attenuated space, the concept of singular linear spaces was firstly introduced in [1]. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular linear space over finite fields, and exhibit their disjunct properties. Moreover, we show that the new construction gives better ratio of efficiency than the former ones under certain conditions. Finally, the paper gives a brief introduction about the relationship between the columns (rows) of the matrix and the related parameters.

A map is unicursal if all its vertices are even-valent except two odd-valent vertices. This paper investigates the enumeration of rooted nonseparable unicursal planar maps and provides two functional equations satisfied by its generating functions with the number of nonrooted vertices, the number of inner faces (or the number of edges) and the valencies of the two odd vertices of maps as parameters.

Let \(\sigma_k(G)\) denote the minimum degree sum of \(k\) independent vertices of a graph \(G\). A spanning tree with at most \(3\) leaves is called a spanning \(3\)-ended tree. In this paper, we prove that for any \(k\)-connected claw-free graph \(G\) with \(|G| = n\), if \(\sigma_{k+3}(G) \geq n – k\), then \(G\) contains a spanning \(3\)-ended tree.

As a promotion of the channel assignment problem, an \(L(1,1,1)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(1\), and the difference between labels of vertices that are distance two and three apart is at least \(1\). About \(10\) years ago, many mathematicians considered colorings (proper, general, total or from lists) such that vertices (all or adjacent) are distinguished either by sets or multisets or sums. In this paper, we will study \(L(1,1,1)\)-labeling-number and \(L(1,1)\)-edge-labeling-number of the edge-path-replacement. From this, we will consider the total-neighbor-distinguishing coloring and the neighbor-distinguishing coloring of the edge-multiplicity-paths-replacements, give a reference for the conjectures: \(\text{tndis-}_\Sigma(G) \leq \Delta + 3\), \(\text{ndi}_\Sigma(G) \leq \Delta + 2\), and \(\text{tndi}_S(G) \leq \Delta + 3\) for the edge-multiplicity-paths-replacements \(G(rP_k)\) with \(k \geq 3\) and \(r \geq 1\).

A \(T\)-shape tree is a tree with exactly one of its vertices having maximal degree \(3\). In this paper, we consider a class of tricyclic graphs which is obtained from a \(T\)-shape tree by attaching three identical odd cycles \(C_ks\) to three vertices of degree \(1\) of the \(T\)-shape tree, respectively, where \(k \geq 3\) is odd. It is shown that such graphs are determined by their adjacency spectrum.

In this paper, we have proved that if a contraction critical \(8\)-connected graph \(G\) has no vertices of degree \(8\), then for every vertex \(x\) of \(G\), either \(x\) is adjacent to a vertex of degree \(9\), or there are at least \(4\) vertices of degree \(9\) such that every one of them is at distance \(2\) from \(x\).