An \(h\)-edge-coloring (block-coloring) of type \(s\) of a graph \(G\) is an assignment of \(h\) colors to the edges (blocks) of \(G\) such that for every vertex \(x\) of \(G\), the edges (blocks) incident with \(x\) are colored with \(s\) colors. For every color \(i\), \(\xi_{x,i}\) (\(\mathcal{B}_{x,i}\)) denotes the set of all edges (blocks) incident with \(x\) and colored by \(i\). An \(h\)-edge-coloring (\(h\)-block-coloring) of type \(s\) is equitable if for every vertex \(x\) and for colors \(i\), \(j\), \(||\xi_{x,i}| – |\xi_{x,j}|| \leq 1\) (\(||\mathcal{B}_{x,i}| – |\mathcal{B}_{x,j}|| \leq 1\)). In this paper, we study the existence of \(h\)-edge-colorings of type \(s = 2,3\) of \(K_t\) and then show that the solution of this problem induces the solution of the existence of a \(C_4\)-\(_tK_2\)-design having an equitable \(h\)-block-coloring of type \(s = 2,3\).

G. Chartrand et al. [3] define a graph \(G\) without isolated vertices to be the least common multiple (lcm) of two graphs \(G_1\) and \(G_2\) if \(G\) is a graph of minimum size such that \(G\) is both \(G_1\)-decomposable and \(G_2\)-decomposable. A bi-star \(B_{m,n}\) is a caterpillar with spine length one. In this paper, we discuss a good lower bound for \(lcm(B_{m,n}, G)\), where \(G\) is a simple graph. We also investigate \(lcm(B_{m,n}, rK_2)\) and provide a good lower bound and an appropriate upper bound for \(lcm(B_{m,n}, P_{r+1})\) for all \(m \geq 1\), \(n \geq 1\), and \(r \geq 1\).

A path in an edge-colored graph is said to be a rainbow path if no two edges on the path share the same color. An edge-colored graph \(G\) is rainbow connected if there exists a \(u-v\) rainbow path for any two vertices \(u\) and \(v\) in \(G\). The rainbow connection number of a graph \(G\), denoted by \(rc(G)\), is the smallest number of colors that are needed in order to make \(G\) rainbow connected. For any two vertices \(u\) and \(v\) of \(G\), a rainbow \(u-v\) geodesic in \(G\) is a rainbow \(u\)–\(v\) path of length \(d(u,v)\), where \(d(u,v)\) is the distance between \(u\) and \(v\). The graph \(G\) is strongly rainbow connected if there exists a rainbow \(u-v\) geodesic for any two vertices \(u\) and \(v\) in \(G\). The strong rainbow connection number of \(G\), denoted by \(src(G)\), is the smallest number of colors that are needed in order to make \(G\) strongly rainbow connected.

In this paper, we determine the precise (strong) rainbow connection numbers of ladders and Möbius ladders. Let \(p\) be an odd prime; we show the (strong) rainbow connection numbers of Cayley graphs on the dihedral group \(D_{2p}\) of order \(2p\) and the cyclic group \(\mathbb{Z}_{2p}\) of order \(2p\). In particular, an open problem posed by Li et al. in [8] is solved.

Given a collection of graphs \(\mathcal{H}\), an \(\mathcal{H}\)-decomposition of \(\lambda K_v\) is a decomposition of the edges of \(\lambda K_v\) into isomorphic copies of graphs in \(\mathcal{H}\). A kite is a triangle with a tail consisting of a single edge. In this paper, we investigate the decomposition problem when \(\mathcal{H}\) is the set containing a kite and a \(4\)-cycle, that is, this paper gives a complete solution to the problem of decomposing \(\lambda K_v\) into \(r\) kites and \(s\) \(4\)-cycles for every admissible values of \(v\), \(r,\lambda\), and \(s\).

A \(b\)-coloring of a graph \(G\) with \(k\) colors is a proper coloring of \(G\) using \(k\) colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer \(k\) for which \(G\) has a \(b\)-coloring using \(k\) colors is the \(b\)-chromatic number \(\beta(G)\) of \(G\). The \(b\)-spectrum \(\mathcal{S}_b(G)\) of a graph \(G\) is the set of positive integers \(k\), \(\chi(G) \leq k \leq b(G)\), for which \(G\) has a \(b\)-coloring using \(k\) colors. A graph \(G\) is \(b\)-continuous if \(\mathcal{S}_b(G) = \{\chi(G), \ldots, b(G)\}\). It is known that for any two graphs \(G\) and \(H\), \(b(G \Box H) \geq \max\{b(G), b(H)\}\), where \(\Box\) stands for the Cartesian product. In this paper, we determine some families of graphs \(G\) and \(H\) for which \(b(G \Box H) \geq b(G) + b(H) – 1\). Further, we show that if \(O_k,i=1,2,\ldots,n\) are odd graphs with \(k_i \geq 4\) for each \(i\), then \(O_{k_1} \Box O_{k_2} \Box \ldots \Box O_{k_n}\) is \(b\)-continuous and \(b(O_{k_1} \Box O_{k_2} \Box \ldots \Box O_{k_n}) = 1 + \sum\limits_{i=1}^{n} k_i\).

We study the number of elements \(x\) and \(y\) of a finite group \(G\) such that \(x \otimes y = 1_{G \oplus G}\) in the nonabelian tensor square \(G \otimes G\) of \(G\). This number, divided by \(|G|^2\), is called the tensor degree of \(G\) and has connections with the exterior degree, introduced a few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra \(39 (2011), 335–343\)]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.

For a (molecular) graph \(G\), the general sum-connectivity index \(\chi_\alpha(G)\) is defined as the sum of the weights \((d_u + d_v)^\alpha\) of all edges \(uv\) of \(G\), where \(d_u\) (or \(d_v\)) denotes the degree of a vertex \(u\) (or \(v\)) in \(G\) and \(\alpha\) is an arbitrary real number. In this paper, we give an efficient formula for computing the general sum-connectivity index of polyomino chains and characterize the extremal polyomino chains with respect to this index, which generalizes one of the main results in [Z. Yarahmadi, A. Ashrafi, S. Moradi, Extremal polyomino chains with respect to Zagreb indices, Appl. Math. Lett. 25 (2012): 166-171].

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.