The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping \(f\) is called a sum labelling of a graph \(G(V, E)\) if it is an injection from \(V\) to a set of positive integers such that \(uv \in E\) if and only if there exists a vertex \(w \in V\) such that \(f(w) = f(x) + f(y)\). In this case, \(w\) is called a working vertex. If \(f\) is a sum labelling of \(G\) with \(r\) isolated vertices, for some nonnegative integer \(r\), and \(G\) contains no working vertex, \(f\) is defined as an exclusive sum labelling of the graph \(G\) by M. Miller et al. in paper [2]. The least possible number \(r\) of such isolated vertices is called the exclusive sum number of \(G\), denoted by \(\epsilon(G)\). If \(\epsilon(G) = \Delta(G)\), the labelling is called \(\Delta\)-optimum exclusive sum labelling and the graph is said to be \(\Delta\)-optimum summable, where \(\Delta = \Delta(G)\) denotes the maximum degree of vertices in \(G\). By using the notion of \(\Delta\)-optimum forbidden subgraph of a graph, the exclusive sum numbers of crown \(C_n \odot K_1\) and \((C_n \odot K_1)\) are given in this paper. Some \(\Delta\)-optimum forbidden subgraphs of trees are studied, and we prove that for any integer \(\Delta \geq 3\), there exist trees not \(\Delta\)-optimum summable. A nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.

In this paper we obtain the Fibonacci length of amalgamated free products having as factors dihedral groups.

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.