Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for every \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament \(T\) of cardinality \(\geq 5\) such that for any vertex \(x\) of \(T\), the tournament \(T – x\) is decomposable. The critical tournaments are of odd cardinality and for all \(n \geq 2\) there are exactly three critical tournaments on \(2n + 1\) vertices denoted by \(T_{2n+1}\), \(U_{2n+1}\), and \(W_{2n+1}\). The tournaments \(T_5\), \(U_5\), and \(W_5\) are the unique indecomposable tournaments on 5 vertices. We say that a tournament \(T\) embeds into a tournament \(T’\) when \(T\) is isomorphic to a subtournament of \(T’\). A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and \(T_5\) embed into an indecomposable tournament \(T\), then \(W_5\) and \(U_5\) embed into \(T’\). To conclude, we prove the following: given an indecomposable tournament \(T\) with \(|V(T)| \geq 7\), \(T\) is critical if and only if only one of the tournaments \(T_7\), \(U_7\), or \(W_7\) embeds into \(T\).

Let \(\lambda K_{m,n}\) be a complete bipartite multigraph with two partite sets having \(m\) and \(n\) vertices, respectively. A \(K_{p,q}\)-factorization of \(\lambda K_{m,n}\) is a set of edge-disjoint \(K_{p,q}\)-factors of \(\lambda K_{m,n}\) which is a partition of the set of edges of \(\lambda K_{m,n}\). When \(\lambda = 1\), Martin, in paper [Complete bipartite factorisations by complete bipartite graphs, Discrete Math., \(167/168 (1997), 461-480]\), gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will give similar necessary conditions for \(\lambda K_{m,n}\) to have a \(K_{p,q}\)-factorization, and prove the necessary conditions are always sufficient in many cases.

In this paper, we determine upper and lower bounds for the number of independent sets in a bicyclic graph in terms of its order. This
gives an upper bound for the total number of independent sets in a connected graph which contains at least two cycles. In each case, we characterize the extremal graphs.

Let \(G\) be a connected graph of order \(n\). Denote \(p_u(G)\) the order of a longest path starting at vertex \(u\) in \(G\). In this paper, we prove that if \(G\) has more than \(t\binom{k}{2} + \binom{p+1}{2} + (n-k-1)\) edges, where \(k \geq 2\), \(n = t(k-1) + p + 1\), \(t \geq 0\) and \(0 \leq p \leq k-1\), then \(p_u(G) > k\) for each vertex \(u\) in \(G\). By this result, we give an alternative proof of a result obtained by P. Wang et al. that if \(G\) is a 2-connected graph on \(n\) vertices and with more than \(t\binom{k-2}{2} + \binom{p}{2} + (2n – 3)\) edges, where \(k \geq 3\), \(n-2 = t(k-2) + p\), \(t \geq 0\) and \(0 \leq p \leq k-2\), then each edge of \(G\) lies on a cycle of order more than \(k\).

In this paper, we give some identities involving the harmonic numbers and the inverses of binomial coefficients.

In this paper, a new efficient computational algorithm is presented for solving cyclic heptadiagonal linear systems based on using the heptadiagonal linear solver and Sherman–Morrison–Woodbury formula. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Two numerical examples are presented for illustration.

Let \(G\) be a graph, and let \(a, b\), and \(k\) be nonnegative integers with \(0 \leq a \leq b\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, we prove that if \(\delta(G) \geq a + k\) and \(\alpha(G) \leq \frac{4b(\delta(G)-a+1-1)}{(a+1)^2}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.

A characterization of \(B\)-H-unretractive bipartite graphs is given. Based on this, it is proved that there is no bipartite graph with endotype \(1 \pmod{4}\).

In a graph \(G = (V, E)\), an independent set is a subset \(I\) of \(V(G)\) such that no two vertices in \(I\) are adjacent. A maximum independent set is an independent set of maximum size. 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 study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.

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.