In this paper, we show that the crossing number of the complete multipartite graph \(K_{1,1,3,n}\) is

\[\operatorname{cr}(K_{1,1,3,n}) = 4\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor + \lfloor\frac{3n}{2}\rfloor\]

Our proof depends on Kleitman’s results for the complete bipartite graphs [D. J. Kleitman, The crossing number of \(K_{5,n}\), J. Combin.Theory, \(9 (1970), 315-323\)]..

A near-perfect matching is a matching saturating all but one vertex in a graph. In this note, it is proved that if a graph has a near-perfect matching then it has at least two, moreover, a concise structure construction for all graphs with exactly two near-perfect matchings is given. We also prove that every connected claw-free graph \(G\) of odd order \(n\) (\(n \geq 3\)) has at least \(\frac{n+1}{2}\) near-perfect matchings which miss different vertices of \(G\).

In this paper, we introduce some contractive conditions of Meir-Keeler type for a pair of mappings, called MK-pair and L-pair, in the framework of cone metric spaces. We prove theorems which assure the existence and uniqueness of common fixed points for MK-pairs and L-pairs. As an application, we obtain a result on the common fixed point of a p-MK-pair, a mapping, and a multifunction in complete cone metric spaces. These results extend and generalize well-known comparable results in the literature.

Four new combinatorial identities involving certain generalized \(F\)-partition functions and \(n\)-colour partition functions are proved bijectively. This leads to new combinatorial interpretations of four mock theta functions of S.Ramanujan.

le of an edge-coloured graph \(G^*\) such that there is no finite integer \(n\) for which it is possible to decompose \(rK_n^*\) into edge-disjoint colour-identical copies of \(G^*\). We investigate the problem of determining precisely when an edge-coloured graph \(G^*\) with \(r\) colours admits a \(G^*\)-decomposition of \(rK_n^*\), for some finite \(n\). We also investigate conditions under which any partial edge-coloured \(G^*\)-decomposition of \(rK_n^*\) has a finite embedding.

Let \(G\) be a connected graph, and let \(d(u,v)\) denote the distance between vertices \(u\) and \(v\) in \(G\). For any cyclic ordering \(\pi\) of \(V(G)\), let \(\pi = (v_1, v_2, \ldots, v_n, v_{n+1} = v_1)\), and let \(d(\pi) = \sum\limits_{i=1}^n d(v_i, v_{i+1})\). The set of possible values of \(d(\pi)\) of all cyclic orderings \(\pi\) of \(V(G)\) is called the Hamiltonian spectrum of \(G\). We determine the Hamiltonian spectrum for any tree.

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f : V(D) \rightarrow \{0, 1, \ldots, |V|\}\) such that the induced function \(f’ : E(D) \rightarrow \{1, 2, \ldots, |V|\}\) which is defined by \(f'(u,v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u,v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of digraph \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of digraph \(D(V,E)\). In this paper, we discuss the gracefulness of the digraph \(n-\vec{C}_m\) and prove that the digraph \(n-\vec{C}_{17}\) is graceful for even \(n\).

Candelabra quadruple systems, which are usually denoted by \(\text{CQS}(g^n : s)\), can be used in recursive constructions to build Steiner quadruple systems. In this paper, we introduce some necessary conditions for the existence of a \(\text{CQS}(g^n : s)\) and settle the existence when \(n = 4,5\) and \(g\) is even. Finally, we get that for any \(n \in \{n \geq 3: n \equiv 2,6 \pmod{12}\) and \(n \neq 8\}\), there exists a \(\text{CQS}(g^n : s)\) for all \(g \equiv 0 \pmod{6}\), \(s \equiv 0 \pmod{2}\) and \(0 \leq s \leq g\).

Let \(G\) be a non-abelian group and let \(Z(G)\) be the center of \(G\). Associate with \(G\) a graph \(\Gamma_G\) as follows: Take \(G\setminus Z(G)\) as vertices of \(\Gamma_G\) and join two distinct vertices \(x\) and \(y\) whenever \(xy \neq yx\). Graph \(\Gamma_G\) is called the non-commuting graph of \(G\) and many of graph theoretical properties of \(\Gamma_G\) have been studied. In this paper, we study some metric graph properties of \(\Gamma_G\).

For integers \(p\), \(q\), \(s\) with \(p \geq q \geq 2\) and \(s \geq 0\), let \(\mathcal{K}_{2}^{-s}(p,q)\) denote the set of \(2\)-connected bipartite graphs which can be obtained from the complete bipartite graph \(K_{p,q}\) by deleting a set of \(s\) edges. F.M.Dong et al. (Discrete Math. vol.\(224 (2000) 107-124\)) proved that for any graph \(G \in \mathcal{K}_{2}^{-s}(p,q)\) with \(p \geq q \geq 3\) and \(0 \leq s \leq \min\{4, q-1\}\), then \(G\) is chromatically unique. In \([13]\), we extended this result to \(s = 5\) and \(s = 6\). In this paper, we consider the case when \(s = 7\).