Let \(G\) be a \(k\)-connected graph and let \(F\) be the simple graph obtained from \(G\) by removing the edge \(xy\) and identifying \(x\) and \(y\) in such a way that the resulting vertex is incident to all those edges (other than \(xy\)) which are originally incident to \(x\) or \(y\). We say that \(e\) is contractible if \(F\) is \(k\)-connected. A bowtie is the graph consisting of two triangles with exactly one vertex in common. We prove that if a \(k\)-connected graph \(G\) (\(k \geq 4\)) has no contractible edge, then there exists a bowtie in \(G\).

We prove that the number of nonisomorphic minimal \(2\)-colorings of the edges of \(K_{4n+3}\) is at least \(2n\) less than the number of nonisomorphic minimal \(2\)-colorings of the edges of \(K_{4n+2}\), where \(n\) is a nonnegative integer. Harary explicitly gave all the nonisomorphic minimal \(2\)-colorings of the edges of \(K_6\). In this paper, we give all the nonisomorphic minimal \(2\)-colorings of the edges of \(K_7\).

We restate a recent improvement of the inclusion-exclusion principle in terms of valuations on distributive lattices and present a completely new proof of the result. Moreover, we establish set-theoretic identities and logical equivalences of inclusion-exclusion type, which have not been considered before.

Let \(\delta(G)\) denote the minimum degree of a graph \(G\). We prove that for \(t \geq 4\) and \(k \geq 2\), a graph \(G\) of order at least \((t + 1)k + 2t^2 – 4t + 2\) with \(\delta(G) \geq k+t- 1\) contains \(k\) pairwise vertex-disjoint \(K_{1,t}\)’s.

In this paper, we construct a squag \(SQG(3n)\) of cardinality \(3n\) that contains three given arbitrary squags \(SQG(n)\)s as disjoint subquags. Accordingly, we can construct a subdirectly irreducible squag \(SQG(3n)\), for each \(n \geq 7\), with \(n \equiv 0, 3 \pmod{6}\). Also, we want to review the shape of the congruence lattice of non-simple squags \(SQG(n)\) for some \(n\) and to give a classification of the class of all \(SQG(21)\)s and the class of all \(SQG(27)\)s according to the shape of its congruence lattice. \(SQG(21)\)s are classified into three classes and \(SQG(27)\)s are classified into four classes. The construction of \(SQG(3n)\), which is given in this paper, helps us to construct examples of each class of both \(SQG(21)\)s and \(SQG(27)\)s.

We show how to produce algebraically a complete orthogonal set of Latin squares from a left quasifield and how to generate algebraically a maximal set of self-orthogonal Latin squares from a left nearfield.

A \((k;g)\)-graph is a \(k\)-regular graph with girth \(g\). A \((k; g)\)-cage is a \((k; g)\)-graph with the least possible number of vertices. In this paper, we prove that all \((4; g)\)-cages are \(4\)-connected, a special case of the conjecture about \((k; g)\)-cages’ connectivity made by H.L. Fu \(et\; al [1]\).

A set \(S\) of vertices of a graph \(G\) is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). Let \(G\) be a spanning subgraph of \(K_{s,s}\), and let \(H\) be the complement of \(G\) relative to \(K_{s,s}\); that is, \(K_{s,s} = G \oplus H\) is a factorization of \(K_{s,s}\). The graph \(G\) is \(k\)-critical relative to \(K_{s,s}\) if \(\gamma_t(G) = k\) and \(\gamma_t(G + e) < k\) for all \(e \in E(H)\). We study \(k_t\)-critical graphs relative to \(K_{s,s}\) for small values of \(k\). In particular, we characterize the \(3\)-critical and \(4_t\)-critical graphs.

Let \(S\) be a nonempty subset of the cyclic group \(\mathbb{Z}_p\), where \(p\) is an odd prime. Denote the \(n\)-fold sum of \(S\) as \(n..S\). That is,\(n..S = \{s_1 + \cdots + s_n \mid s_1, \ldots, s_n \in S\}.\) We say that \(S\) is an \((n, 0)\)-set if \(0 \notin n..S\). Let \(k, s\) be integers with \(k \geq 2\) such that \(p-1 = ks\). In this paper, we determine the number of \((k, 0)\)-sets of \(\mathbb{Z}_p\) which are in arithmetic progression and show explicitly the forms taken by those \((k, 0)\)-sets which achieve the maximum cardinality.

In this paper, necessary and sufficient conditions are given for the existence of extended \(5\)-cycle systems of order \(n\) which have \(r\) idempotent elements.