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.

A total dominating set of a graph \(G\) with no isolated vertex is a set \(S\) of vertices of \(G\) such that every vertex is adjacent to a vertex in \(S\). The total domination number of \(G\) is the minimum cardinality of a total dominating set in \(G\). In this paper, we present several upper bounds on the total domination number in terms of the minimum degree, diameter, girth, and order.

We denote by \((p, q)\)-graph \(G\) a graph with \(p\) vertices and \(q\) edges. An edge-magic total (EMT) labeling on a \((p,q)\)-graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow [1,2,\ldots,p+q]\) with the property that, for each edge \(xy\) of \(G\), \(\lambda(x) + \lambda(xy) + \lambda(y) = k\), for a fixed positive integer \(k\). Moreover, \(\lambda\) is a super edge-magic total labeling (SEMT) if it has the property that \(\lambda(V(G)) = \{1, 2,\ldots,p\}\). A \((p,q)\)-graph \(G\) is called EMT (SEMT) if there exists an EMT (SEMT) labeling of \(G\). In this paper, we propose further properties of the SEMT graph. Based on these conditions, we will give new theorems on how to construct new SEMT (bigger) graphs from old (smaller) ones. We also give the SEMT labeling of \(P_n \cup P_{n+m}\) for possible magic constants \(k\) and \(m = 1, 2\),or \(3\).

A Kirkman packing design \(KPD({w, s^*, t^*}, v)\) is a Kirkman packing with maximum possible number of parallel classes, such that each parallel class contains one block of size \(s\), one block of size \(t\) and all other blocks of size \(w\). A \((k, w)\)-threshold scheme is a way of distributing partial information (shadows) to \(w\) participants, so that any \(k\) of them can determine a key easily, but no subset of fewer than \(k\) participants can calculate the key. In this paper, the existence of a \(KPD({3, 4^*, 5^*}, v)\) is established for every \(v \equiv 3 \pmod{6}\) with \(v \geq 51\). As its consequence, some new \((2, w)\)-threshold schemes have been obtained.