We conjectured in \([3]\) that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the \(3\)-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete \(k\)-partite, and matrogenic cyclic graphs are weakly split.

Let \((X,\mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition and let \(G_i\), \(i = 1,\ldots,\mu\), be nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i | B \in \mathcal{B}\}\), where \(\mathcal{B_i}\) is a subgraph of \(B\)  isomorphic to \(G_i\). A \(\{G_1,G_2,\ldots,G_\mu\}\)-metamorphosis of \((X,\mathcal{B})\) is a rearrangement, for each \(i=1,\ldots,\mu\), of the edges of \(\bigcup_{B\in B}(E(B)\setminus\mathcal{B}_i))\) into a family \(\mathcal{F}_i\) of copies of \(G_i\) with a leave \(L_i\), such that \((X,\mathcal{B}_i \cup \mathcal{F}_i,L_i)\) is a maximum packing of \(\lambda H\) with copies of \(G_i\). In this paper, we give a complete answer to the existence problem of an \(S_\lambda(2,4,7)\) having a \(\{C_4, K_3 + e\}\)-metamorphosis.

For a positive integer \(m\), where \(1 \leq m \leq n\), the \(m\)-competition index (generalized competition index) of a primitive digraph \(D\) of order \(n\) is the smallest positive integer \(k\) such that for every pair of vertices \(x\) and \(y\), there exist \(m\) distinct vertices \(v_1, v_2, \ldots, v_m\) such that there exist walks of length \(k\) from \(x\) to \(v_i\) and from \(y\) to \(v_i\) for \(1 \leq i \leq m\). In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.

We give a new combinatorial bijection between a certain set of balanced modular tableaux of Gusein-Zade, Luengo, and Melle-Hernandez and \(k\)-ribbon shapes. In addition, we also use the Schensted algorithm for the rim hook tableaux of Stanton and White to write down an explicit generating function for these balanced modular tableaux.

A \((k;g)\)-cage is a graph with the minimum order among all \(k\)-regular graphs with girth \(g\). As a special family of graphs, \((k;g)\)-cages have a number of interesting properties. In this paper, we investigate various properties of cages, e.g., connectivity, the density of shortest cycles, bricks and braces.

Let \(G = (V, E)\) be a digraph with \(n\) vertices and \(m\) arcs without loops and multiarcs, \(V = \{v_1, v_2, \ldots, v_n\}\). Denote the outdegree and average \(2\)-outdegree of the vertex \(v_i\) by \(d^+_i\) and \(m^+_i\), respectively. Let \(A(G)\) be the adjacency matrix and \(D(G) = \text{diag}(d^+_1, d^+_2, \ldots, d^+_n)\) be the diagonal matrix with outdegrees of the vertices of the digraph \(G\). Then we call \(Q(G) = D(G) + A(G)\) the signless Laplacian matrix of \(G\). In this paper, we obtain some upper and lower bounds for the spectral radius of \(Q(G)\), which is called the signless Laplacian spectral radius of \(G\). We also show that some bounds involving outdegrees and the average \(2\)-outdegrees of the vertices of \(G\) can be obtained from our bounds.

Lee and Kong conjecture that if \(n \geq 1\) is an odd number, then \(St(a_1, a_0, \ldots, a_n)\) would be super edge-magic, and meanwhile they proved that the following graphs are super edge-magic: \(St(m,n)\) (\(n = 0 \mod (m+1)\)), \(St(1,k,n)\) (\(k = 1,2\) or \(n\)), \(St(2, k,n)\) (\(k = 2,3\)), \(St(1,1,k,n)\) (\(k = 2,3\)), \(St(k,2,2,n)\) (\(k = 1,2\)). In this paper, the conjecture is further discussed and it is proved that \(St(1,m,n)\), \(St(3,m,m+1)\), \(St(n,n+1,n+2)\) are super edge-magic, and under some conditions \(St(a_1, a_2, \ldots, a_{2n+1})\); \(St(a_1, a_2, \ldots, a_{4n+1})\), \(St(a_1, a_2, \ldots, a_{4n+3})\) are also super edge-magic.

We determine all connected odd graceful graphs of order \(\leq 6\). We show that if \(G\) is an odd graceful graph, then \(G \cup K_{m,n}\) is odd graceful for all \(m, n \geq 1\). We give an analogous statement to the graceful graphs statement, and we show that some families of graphs are odd graceful.

In this paper, we provide a method to obtain the lower bound on the number of distinct maximum genus embeddings of the complete bipartite graph \(K_{n,n}\) (\(n\) is an odd number), which, in some sense, improves the results of S. Stahl and H. Ren.

For positive integer \(n\), let \(f_3(n)\) be the least upper bound of the sums of the lengths of the sides of \(n\) cubes packed into a unit cube \(C\) in three dimensions in such a way that the smaller cubes have sides parallel to those of \(C\). In this paper, we improve the lower bound of \(f_3(n)\).