We give some relationships among the intersection numbers of a distance-regular graph \(\Gamma\) which contains a circuit \((u_1,u_2,u_3,u_4)\) with \(\partial(u_1,u_2) = 1\) and \(\partial(u_2,u_4) = 2\). As an application, we obtain an upper bound of the diameter of \(\Gamma\) when \(k \geq 2b_1\).

We extend results concerning orthogonal edge labeling of constant weight Gray codes. For positive integers \(n\) and \(r\) with \(n > r\), let \(G_{n,r}\) be the graph whose vertices are the \(r\)-sets of \(\{1, \ldots, n\}\), with \(r\)-sets adjacent if they intersect in \(r-1\) elements. The graph \(G_{n,r}\) is Hamiltonian; Hamiltonian cycles of \(G_{n,r}\) are early examples of error-correcting codes, where they came to be known as constant weight Gray codes.

An \(r\)-set \(A\) and a partition \(\pi\) of weight \(r\) are said to be orthogonal if every block of \(\pi\) meets \(A\) in exactly one element. Given a class \(P\) of weight \(r\) partitions of \(X_n\), one would like to know if there exists a \(G_{n,r}\) Hamiltonian cycle \(A_1 A_2 \ldots A_{\binom{n}{r}}\) whose edges admit a labeling \(A_1\pi_1 A_2 \ldots A_{\binom{n}{r}}\pi_{\binom{n}{r}}\) by distinct partitions from \(\mathcal{P}\), such that a partition label of an edge is orthogonal to the vertices that comprise the edge. The answer provides non-trivial information about Hamiltonian cycles in \(G_{n,r}\) and has application to questions pertaining to the efficient generation of finite semigroups.

Let \(r\) be a partition of \(m\) as a sum of \(r\) positive integers. We let \(r\) also refer to the set of all partitions of \(X_n\) whose block sizes comprise the partition \(r\). J. Lehel and the first author have conjectured that for \(n \geq 6\) and partition type \(\pi\) of \(\{1, \ldots, n\}\) of weight \(r\) partitions, there exists a \(r\)-labeled Hamiltonian cycle in \(G_{n,r}\).

In the present paper, for \(n = s + r\), we prove that there exist Hamiltonian cycles in \(G_{n,r}\) which admit orthogonal labelings by the partition types which have \(s\) blocks of size two and \(r – s\) blocks of size one, thereby extending a result of J. Lehel and the first author and completing the work on the conjecture for all partition types with blocks of size at most two.

For distinct vertices \(u\) and \(v\) of a nontrivial connected graph \(G\), the detour distance \(D(u,v)\) between \(u\) and \(v\) is the length of a longest \(u-v\) path in \(G\). For a vertex \(v \in V(G)\), define \(D^-(v) = \min\{D(u,v) : u \in V(G) – \{v\}\}\). A vertex \(u (\neq v)\) is called a detour neighbor of \(v\) if \(D(u,v) = D^-(v)\). A vertex \(v\) is said to detour dominate a vertex \(u\) if \(u = v\) or \(u\) is a detour neighbor of \(v\). A set \(S\) of vertices of \(G\) is called a detour dominating set if every vertex of \(G\) is detour dominated by some vertex in \(S\). A detour dominating set of \(G\) of minimum cardinality is a minimum detour dominating set and this cardinality is the detour domination number \(\gamma_D(G)\). We show that if \(G\) is a connected graph of order \(n \geq 3\), then \(\gamma_D(G) \leq n-2\). Moreover, for every pair \(k,n\) of integers with \(1 \leq k \leq n-2\), there exists a connected graph \(G\) of order \(n\) such that \(\gamma_D(G) = k\). It is also shown that for each pair \(a,b\) of positive integers, there is a connected graph \(G\) with domination number \(\gamma(G) = a\) and \(\gamma_D(G) = b\).

We let \(A(n)\) equal the number of \(n \times n\) alternating sign matrices. From the work of a variety of sources, we know that

\[A(n) = \prod\limits_{t=0}^{n-1} \frac{(3l+1)!}{(n+l)!}\]

We find an efficient method of determining \(ord_p(A(n))\), the highest power of \(p\) which divides \(A(n)\), for a given prime \(p\) and positive integer \(n\), which allows us to efficiently compute the prime factorization of \(A(n)\). We then use our method to show that for any nonnegative integer \(k\), and for any prime \(p > 3\), there are infinitely many positive integers \(n\) such that \(ord_p(A(n)) = k\). We show a similar but weaker theorem for the prime \(p = 3\), and note that the opposite is true for \(p = 2\).

We survey the status of minimal coverings of pairs with block sizes two, three, and four when \(\lambda = 1\), that is, all pairs from a \(v\)-set are covered exactly once. Then we provide a complete solution for the case \(\lambda = 2\).

We derive an alternative rule for generating uniform step magic squares. The compatibility conditions for the proposed rule are simpler than the analogous conditions for the classical uniform step rule. We exploit this fact to enumerate all uniform-step magic squares of every given odd order. Our main result states that if \(p = \prod_{i=1}^l q_i^{r_i}\) is the prime factorization of a positive odd number \(p\), then there exist \(\kappa(p) =\prod _{i=1}^l \kappa(q_i^{r_i})\) uniform step magic squares of order \(p\), where
\(\kappa (q_i^{r_i})=[\tau (q_i^{r_i})]^2-\lambda (q_i^{r_i}),\lambda(q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})^2[2(q_i^{2r_i-1}+1)^2/(q_i+1)^2+q_i^{3r_i-1}(q_i^{r_i}-3q_i^{r_i-1})]\) and \(\tau (q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})(q_i^{2r_i+1}-2q_i^{2r_i}-q_i^{2r_i-1}+2)/(q_i+1)\) for \(i=1,\ldots,l\)

We show that for a cubic graph on \(n\) nodes, the size of the dominating set found by the greedy algorithm is at most \(\frac{4}{9}n\), and that this bound is tight.

For a standard tableau \(T\) of shape \(\lambda \vdash n\), \(maj(T)\) is the sum of \(i\)’s such that \(i+1\) appears in a row strictly below that of \(i\) in \(T\). We consider the \(g\)-polynomial \(f^\lambda(q) = \sum_\tau q^{ maj(T)}\), which appears in many contexts: as a dimension of an irreducible representation of finite general linear group, as a special case of Kostka-Foulkes polynomials, and so on. In this article, we try to understand `maj’ on a standard tableau \(T\) in relation to `inv’ on a multiset permutation (or a permutation of type \(\lambda\)). We construct an injective map from the set of standard tableaux to the set of permutations of type \(\lambda\) (increasing in each block) such that the `maj’ of the tableau is the `maj’ of the corresponding permutation when \(\lambda\) is a two-part partition. We believe that this helps to understand irreducible unipotent representations of finite general linear groups.

Many results about outer-embeddings (graphs having all their vertices in the same face) have been obtained recently in topological graph theory in recent times. In this paper, we deal with some difficulties appearing in the study of such embeddings. Particularly, we propose several problems concerning outer-embeddings in pseudosurfaces and we prove that two of them are NP-complete.

We also describe some properties about lists of forbidden minors for outer-embeddings in certain kinds of pseudosurfaces.

Let \(G\) be a graph with integral edge weights. A function \(d: V(G) \to \mathbb{Z}_p\) is called a nowhere \(0 \mod p\) domination function if each \(v \in V\) satisfies \((\sum_{u \in N(v)} w(u,v)d(u))\neq 0 \mod p\), where \(w(u,v)\) denotes the weight of the edge \((u,v)\) and \(N(v)\) is the neighborhood of \(v\). The subset of vertices with \(d(v) \neq 0\) is called a nowhere \(0 \mod p\) dominating set. It is known that every graph has a nowhere \(0 \mod 2\) dominating set. It is known to be false for all other primes \(p\). The problem is open for all odd \(p\) in case all weights are one.

In this paper, we prove that every unicyclic graph (a graph containing at most one cycle) has a nowhere \(0 \mod p\) dominating set for all \(p > 1\). In fact, for trees and cycles with any integral edge weights, or for any other unicyclic graph with no edge weight of \((-1) \mod p\), there is a nowhere \(0 \mod p\) domination function \(d\) taking only \(0-1\) values. This is the first nontrivial infinite family of graphs for which this property is established. We also determine the minimal graphs for which there does not exist a \(0 \mod p\) dominating set for all \(p > 1\) in both the general case and the \(0-1\) case.