In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex \(v\), denoted by \(\text{id}(v)\). In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If \(G\) is a \(2\)-connected graph of order \(n\) such that \(\text{id}(u) + \text{id}(v) \geq n – 1\) for each pair of vertices \(u\) and \(v\) at distance \(2\), then \(G\) is hamiltonian, with some exceptions.

Let \(C_k\) denote a cycle of length \(k\) and let \(S_k\) denote a star with \(k\) edges. For graphs \(F\), \(G\), and \(H\), a \((G, H)\)-multidecomposition of \(F\) is a partition of the edge set of \(F\) into copies of \(G\) and copies of \(H\) with at least one copy of \(G\) and at least one copy of \(H\). In this paper, necessary and sufficient conditions for the existence of the \((C_k, S_k)\)-multidecomposition of a complete bipartite graph are given.

This paper investigates the number of boundary cubic inner-forest maps and presents some formulae for such maps with the size (number of edges) and the valency of the root-face as two parameters. Further, by duality, some corresponding results for rooted outer-planar maps are obtained. It is also an answer to the open problem in \([15]\) and corrects the result on boundary cubic inner-tree maps in \([15]\).

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.