Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that the graphs \(C_n^{(t)}\) are graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 4k\). In this paper, the conjecture is shown to be true for \(n = 7\).

It is well known that a linear code over a finite field with the systematic generator matrix \([I | P]\) is MDS (Maximum Distance Separable) if and only if every square submatrix of \(P\) is nonsingular. In this correspondence, we obtain a similar characterization for the class of Near-MDS codes in terms of the submatrices of \(P\).

In this paper, we define the signed total domatic number of a graph in an analogous way to that of the fractional domatic number defined by Rall (A fractional version of domatic number. Congr. Numer. \(74 (1990), 100-106)\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A set \(\{f_1,\ldots,f_a\}\) of signed total dominating functions on \(G\) such that \(\sum\limits_{i=1}^a f_i(v) \leq 1\) for each vertex \(v \in V(G)\) is called a signed total dominating family of functions on \(G\). The signed total domatic number of \(G\) is the maximum number of functions in a signed total dominating family of \(G\). In this paper, we investigate the signed total domatic number for special classes of graphs.

Let \(G\) be the product of two directed cycles, let \(Z_a\) be a subgroup of \(Z_a\), and let \(Z_d\) be a subgroup of \(Z_b\). Also, let \(A = \frac{a}{c}\) and \(B = \frac{b}{d}\). We say that \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if there is a spanning connected subgraph of \(G\) that has degree \((2, 2)\) at the vertices of \(Z_c \times Z_d\) and degree \((1, 1)\) everywhere else. We show that the graph \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if and only if there exist positive integers \(m\) and \(n\) such that \(Am + Bn = AB + 1\), \(gcd(m, n) = 1\) or \(2\), and when \(gcd(m, n) = 2\), then \(gcd(dm, cn) = 2\).

In this paper, it is shown that a partial edge-disjoint decomposition of \(K_{n}\) into kites (that is, into copies of \(K_3\) with a pendant edge attached) can be embedded in a complete edge-disjoint decomposition of \(K_{4t+9}\) into kites for all even \(t \geq 2n\). The proof requires first proving another interesting result, a generalization of an embedding result on symmetric latin squares by L. D. Andersen, following a result by A. Cruse.

Let \(T_n\) be the complete binary tree of height \(n\) considered as the Hasse-diagram of a poset with its root \(1_n\) as the maximum element. For a tree or forest \(T\), we count the embeddings of \(T\) into \(T_n\) as posets by the functions \(A(n;T) = |\{S \subseteq T_n : 1_n \in S, S \cong T\}|\), and \(B(n;T) = |\{S \subseteq T_n : 1_n \notin S, S \cong T\}|\). Here we summarize what we know about the ratio \(A(n;T)/B(n;T)\), in case of \(T\) being a chain or an antichain.

In this paper, the concept of clique number of uniform hypergraph is defined and its relationship with circular chromatic number and clique number is studied. For every positive integer \(k, p\) and \(q\), \(2q \leq p\) we construct a \(k\)-uniform hypergraph with small clique number whose circular chromatic number is equal to \(\frac{p}{q}\). We define the concept and study the properties of \(c\)-perfect \(k\)-uniform hypergraphs.

Given a connected graph \(G\) and a subset \(S\) of vertices, the Steiner distance of \(S\) in \(G\) is the minimum number of edges in a tree in \(G\) that contains all of \(S\). Given a positive integer \(m\), let \(\mu_m(G)\) denote the average Steiner distance over all sets \(S\) of \(m\) vertices in \(G\). In particular, \(\mu_2(G)\) is just the average distance of \(G\), often denoted by \(\mu(G)\). Dankelmann, Oellermann, and Swart \([1]\) conjectured that if \(G\) is a connected graph of order \(n\) and \(3 \leq m \leq n\), then \(\frac{\mu_m(G)}{\mu(G)} \geq 3(\frac{m-1}{m+1})\). In this note, we disprove their conjecture by showing that

\[\lim_{m \to \infty} inf \{ \frac{\mu_m(G)}{\mu(G)} :G \text{ is connected and $n(G)\geq m$} \} = 2.\]

Given a simple graph \(G\) on \(n\) vertices, let \(\sigma_2(G)\) be the minimum sum of the degrees of any two non-adjacent vertices. The graph \(G\) is said to be connected if any two distinct vertices may be joined by a path. It is easy to see that if \(\sigma_2(G) \geq n-1\) then \(G\) is not only connected, but we can choose the connecting path to be of size at most two. Ore \([4]\) proved that if \(\sigma_2(G) \geq n+1\) we may always choose this path to cover all the vertices of \(G\). In this paper we extend these results to systems of vertex disjoint paths connecting two vertex \(k\)-sets of \(G\).

A graph with vertex set \(V\) is said to have a prime labeling if its vertices are labelled with distinct integers from \(\{1, 2, \ldots, |V|\}\) such that for each edge \(xy\), the labels assigned to \(x\) and \(y\) are relatively prime. A graph that admits a prime labeling is called a prime graph. It has been conjectured \([1]\) that when \(n\) is a prime integer and \(m < n\), the planar grid \(P_m \times P_n\) is prime. We prove the conjecture and also that \(P_n \times P_n\) is prime when \(n\) is a prime integer.