The following two theorems are proved:
A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a cylinder so that the \(m\) rows go around the cylinder, with one square removed, with the exception of the following boards:

(a) \(n\) is even,

(b) \(m \in \{1,2\}\)

(c) \(m = 4\) and the removed square is in row 2 or 3;

(d) \(m \geq 5\), \(n = 1\), and the removed square is in row 2, 3, …, or \(m-1\).

A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a torus with one square removed except boards with \(m\) and \(n\) both even and \(1 \times 1\),\(1 \times 2\) and \(2 \times 1\) boards.

An independent set \(S\) of a connected graph \(G\) is called a \emph{frame} if \(G – S\) is connected. If \(|S| = k\), then \(S\) is called a \emph{k-frame}. We prove the following theorem.
Let \(k \geq 2\) be an integer, \(G\) be a connected graph with \(V(G) = \{v_1, v_2, \ldots, v_n\}\), and \(\deg_G(u)\) denote the degree of a vertex \(u\). Suppose that for every \(3\)-frame \(S = \{v_a, v_b, v_c\}\) such that \(1 \leq a \leq b \leq c \leq n\), \(\deg_G(v_c) \leq a\), \(\deg_G(v_b) \leq b-1\), and \(\deg_G(v_c) \leq c – 2\), it holds that\[\deg_G(v_a) + \deg_G(v_b) + \deg_G(v_c) – |N(v_a) \cap N(v_b) \cap N(v_c)| \geq |G| – k + 1.\] Then \(G\) has a spanning tree with at most \(k\)-leaves. Moreover, the condition is sharp.
This theorem is a generalization of the results of E. Flandrin, H.A. Jung, and H. Li (Discrete Math. \(90 (1991), 41-52)\) and of A. Kyaw (Australasian Journal of Combinatorics. \(37 (2007), 3-10)\) for traceability.

In this paper, the estimations of maximum genus orientable embeddings of graphs are studied, and an exponential lower bound for such numbers is found. Moreover, such two extremal embeddings (i.e., the maximum genus orientable embedding of the current graph and the minimum genus orientable embedding of the complete graph) are sometimes closely related to each other. As applications, we estimate the number of minimum genus orientable embeddings for the complete graph by estimating the number of maximum genus orientable embeddings for the current graph.

In this article, we characterize for which finite commutative rings \(R\), The zero-divisor graph \(\Gamma(R)\),The line graph \(L(\Gamma(R))\), The complement graph \(\overline{\Gamma(R)}\), and The line graph for the complement graph \(L(\overline{\Gamma(R)})\).

The energy of a graph \(G\) is defined as the sum of the absolute values of all the eigenvalues of the graph. In this paper, we consider the energy of the \(3\)-circulant graphs, and obtain a computation formula, and establish new results for a certain class of circulant graphs. At the same time, we give a conjecture: The largest energy of circulant graphs relates with their components.

In this paper, we present two criteria for a sequence lying along a ray of a combinatorial triangle to be unimodal, and give a correct
proof for the result of Belbachir and Szalay on unimodal rays of the generalized Pascal’s triangle.

In this paper, we introduce the notion of derivation in lattice implication algebra, and consider the properties of derivations in lattice implication algebras. We give an equivalent condition to be a derivation of a lattice implication algebra. Also, we characterize the fixed set \(Fix_d(L)\) and \(Kerd\) by derivations. Moreover, we prove that if \(d\) is a derivation of a lattice implication algebra, every filter \(F\) is \(d\)-invariant.

In this paper, we derive a family of identities on the arbitrary subscripted Fibonacci and Lucas numbers. Furthermore, we construct the tridiagonal and symmetric tridiagonal family of matrices whose determinants form any linear subsequence of the Fibonacci numbers and Lucas numbers. Thus, we give a generalization of the results presented in Nalli and Civciv [A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos, Solitons and Fractals \(2009;40(1):355 .61]\) and Cahill and Narayan [N. D. Cahill, D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, The Fibonacci Quarterly, \(2004;42(1):216–221]\).

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we determine the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.

For a poset \(P = (X, \leq_ P)\), the strict-double-bound graph (\(sDB\)-graph \(sDB(P)\)) is the graph on \(X\) for which vertices \(u\) and \(v\) of \(sDB(P)\) are adjacent if and only if \(u \neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) such that \(x \leq_ P y\) and \(x \leq_P v \leq_P y\). The strict-double-bound number \(\zeta(G)\) of a graph \(G\) is defined as \(\min\{n; G \cup \overline{K}_n \text{ is a strict-double-bound graph}\}\).

We obtain that for a spider \(S_{n,m}\) (\(n,m > 3\)) and a ladder \(L_n\) (\(n \geq 4\)), \(\left\lceil2\sqrt{nm}\right\rceil \leq \zeta(S_{n,m}) \leq n+m\), \(\zeta(S_{n,n}) = 2n\), and \(\left\lceil 2\sqrt{3n+2}\right\rceil \leq \zeta(L_n) \leq 2n\).