Let \(G\) be a finite abelian group with exponent \(n\). Let \(s(G)\) denote the smallest integer \(l\) such that every sequence over \(G\) of length at least \(l\) has a zero-sum subsequence of length \(n\). For \(p\)-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on \(s(G)\), which allows to determine \(s(G)\) precisely in special cases. Our results contain Kemnitz’ conjecture, which was recently proved, as a special case.

Let \(\mathcal{D}\) be a \(2\)-\((v,k,4)\) symmetric design, and \(G\) be a subgroup of the full automorphism group of \(\mathcal{D}\). In this paper, we prove that if \(G \leq {Aut}(\mathcal{D})\) is flag-transitive, point-primitive then \(G\) is of affine or almost simple type. We prove further that if a nontrivial \(2\)-\((v, k, 4)\) symmetric design has a flag-transitive, point-primitive, almost simple automorphism group \(G\), then \(\text{Soc}(G)\) is not a sporadic simple group.

We prove explicit formulas for the rank polynomial and Whitney numbers of the distributive lattice of order ideals of the garland poset, ordered by inclusion.

A semi-double graph is such a connected multi-graph that each multi-edge consists of two edges. If there is at most one loop at each vertex of a semi-double graph, then this graph is called a single-petal graph. In this paper, we obtained that if \(G\) is a connected (resp. \(2\)-edge-connected, \(3\)-edge-connected) simple graph of order \(n\), then \(G\) is upper embeddable if \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-3}{2}\right\rceil\) (resp. \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-2}{3}\right\rceil, d_G(u) + d_G(v) \geq \left\lceil\frac{2n-23}{2}\right\rceil\)) for any two adjacent vertices \(u\) and \(v\) of \(G\). In addition, by means of semi-double graph and single-petal graph, the upper embeddability of multi-graph and pseudograph are also discussed in this paper.

Let \(d(n, k)\) denote the number of derangements (permutations without fixed points) with \(k\) cycles of the set \([n] = \{1, 2, \ldots, n\}\). In this paper, a new explicit expression for \(d(n, k)\) is presented by graph theoretic method, and a concise regular binary tree representation for \(d(n, k)\) is provided.

This paper devotes to the investigation of \(3\)-designs admitting the special projective linear group \(\text{PSL}(2,q)\) as an automorphism group. When \(q \equiv 3 \pmod{4}\), we determine all the possible values of \(\lambda\) in the simple \(3\)-\((q+1, 7, \lambda)\) designs admitting \(\text{PSL}(2,q)\) as an automorphism group.

We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order \(3\). This result is used to give an upper bound of \(2\Delta\) for the strong chromatic number of \(n\) vertex graphs with \(\Delta \geq n/6\).

A partial Latin square \(P\) of order \(n\) is an \(n \times n\) array with entries from the set \(\{1, 2, \ldots, n\}\) such that each symbol is used at most once in each row and at most once in each column. If every cell of the array is filled, we call \(P\) a Latin square. A partial Latin square \(P\) of order \(n\) is said to be avoidable if there exists a Latin square \(L\) of order \(n\) such that \(P\) and \(L\) are disjoint. That is, corresponding cells of \(P\) and \(L\) contain different entries. In this note, we show that, with the trivial exception of the Latin square of order \(1\), every partial Latin square of order congruent to \(1\) modulo \(4\) is avoidable.

For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : 0 \leq i \leq n-1, j = i+1, i+2, \ldots, i+k \pmod{n}\}\). A caterpillar is a tree of order at least three which contains a path such that each vertex not on the path is adjacent to a vertex on the path. Being a connected bipartite graph, a caterpillar is balanced if the two parts of the bipartition of its vertices have equal size; otherwise, it is unbalanced. In this paper, we obtain the necessary and sufficient condition for balanced-caterpillar factorization of crowns. The criterion for unbalanced-caterpillar factorization of crowns is open. We also obtain the necessary and sufficient condition for directed caterpillar factorization of symmetric crowns.

This paper determines that the connectivity of the Cartesian product \(G_1 \square G_2\) of two graphs \(G_1\) and \(G_2\) is equal to \(\min\{\kappa_1v_2 + \kappa_2v_1, \delta_1 + \delta_2 \}\), where \(v_i, \kappa_i\), and \(\delta_i\) are the order, connectivity, and minimum degree of \(G_i\), respectively, for \(i = 1, 2\). Additionally, some necessary and sufficient conditions are given for \(G_1 \square G_2\) to be maximally connected and super-connected.