Let \(G\) be a simple connected graph. For a subset \(S\) of \(V(G)\) with \(|S| = 2n + 1\), let \(\alpha_{(2n+1)}(G,S)\) denote the graph obtained from \(G\) by contracting \(S\) to a single vertex. The graph \(\alpha_{(2n+1)}(G, S)\) is also said to be obtained from \(G\) by an \(\alpha_{(2n+1)}\)-contraction. For pairwise disjoint subsets \(S_1, S_2, \ldots, S_{2n}\) of \(V(G)\), let \(\beta_n(G, S_1, S_2, \ldots, S_{2n})\) denote the graph obtained from \(G\) by contracting each \(S_i\) (\(i = 1, 2, \ldots, 2n\)) to a single vertex respectively. The graph \(\beta_{2n}(G, S_1, S_2, \ldots, S_{2n})\) is also said to be obtained from \(G\) by a \(\beta_{2n}\)-contraction. In the present paper, based on \(\alpha_{(2n+1)}\)-contraction and \(\beta_{2}\)-contraction, some new characterizations for \(n\)-extendable bipartite graphs are given.

A graph \(G\) is quasi-claw-free if it satisfies the property: \(d(x, y) = 2 \Rightarrow\) there exists \(u \in N(x) \cap N(y)\) such that \(N[u] \subseteq N[x] \cup N[y]\). In this paper, we prove that the circumference of a \(2\)-connected quasi-claw-free graph \(G\) on \(n\) vertices is at least \(\min\{3\delta + 2, n\}\) or \(G \in \mathcal{F}\), where \(\mathcal{F}\) is a class of nonhamiltonian graphs of connectivity \(2\). Moreover, we prove that if \(n \leq 40\), then \(G\) is hamiltonian or \(G \in \mathcal{F}\).

Let \(K_{n,n}\) denote the complete bipartite graph with \(n\) vertices in each part. In this paper, it is proved that there is no cyclic \(m\)-cycle system of \(K_{n,n}\) for \(m \equiv 2 \pmod{4}\) and \(n \equiv 2 \pmod{4}\). As a consequence, necessary and sufficient conditions are determined for the existence of cyclic \(m\)-cycle systems of \(K_{n,n}\) for all integers \(m \leq 30\).

We examine a design \(\mathcal{D}\) and a binary code \(C\) constructed from a primitive permutation representation of degree \(2025\) of the sporadic simple group \(M^c L\). We prove that \(\text{Aut}(C) = \text{Aut}(\mathcal{D}) = M^c L\) and determine the weight distribution of the code and that of its dual. In Section \(6\) we show that for a word \(w_i\) of weight \(7\), where \(i \in \{848, 896, 912, 972, 1068, 1100, 1232, 1296\}\) the stabilizer \((M^\circ L)_{w_i}\) is a maximal subgroup of \(M^\circ L\). The words of weight \(1024\) split into two orbits \(C_{(1024)_1}\) and \(C_{(1024)_2}\), respectively. For \(w_i \in C_{(1024)_1}\), we prove that \((M^c L)_{w_i}\) is a maximal subgroup of \(M^c L\).

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v, G, \lambda)\)-PD \(((v, G,\lambda)\)-CD), is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(\mathcal{B}\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we have completely determined the packing number and covering number for the graphs with seven points, seven edges and an even cycle.

In this paper, it is shown that there are exactly \(5\) non-isomorphic abstract ovals of order \(9\), all of them projective. The result has been obtained via an exhaustive search, based on the classification of the \(1\)-factorizations of the complete graph with \(10\) vertices.

A graph \(G\) is said to be \(k\)-degenerate if for every induced subgraph \(H\) of \(G\), \(\delta(H) \leq k\). Clearly, planar graphs without \(3\)-cycles are \(3\)-degenerate. Recently, it was proved that planar graphs without \(5\)-cycles or without \(6\)-cycles are also \(3\)-degenerate. And for every \(k = 4\) or \(k \geq 7\), there exist planar graphs of minimum degree \(4\) without \(k\)-cycles. In this paper, it is shown that each \(C_7\)-free plane graph in which any \(3\)-cycle is adjacent to at most one triangle is \(3\)-degenerate. So it is \(4\)-choosable.

This paper investigates the embedding problem for resolvable group divisible designs with block size \(3\). The necessary and sufficient conditions are determined for all \(\lambda \geq 1\).

We provide combinatorial arguments of some relations between classical Stirling numbers of the second kind and two refinements of these numbers gotten by introducing restrictions to the distances among the elements in each block of a finite set partition.

We provide many new edge-magic and vertex-magic total labelings for the cycles \(C_{nk}\), where \(n \geq 3\) and \(k \geq 3\) are both integers and \(n\) is odd. Our techniques are of interest since known labelings for \(C_{k}\) are used in the construction of those for \(C_{nk}\). This provides significant new evidence for a conjecture on the possible magic constants for edge-magic and vertex-magic cycles.