For given edges \(e_1, e_2 \in E(G)\), a spanning trail of \(G\) with \(e_1\) as the first edge and \(e_2\) as the last edge is called a spanning \((e_1, e_2)\)-trail. In this note, we consider best possible degree conditions to assure the existence of these trails for every pair of edges in a \(3\)-edge-connected graph \(G\).

In this paper, it is proved that an abelian \((351, 126, 45)\)-difference set only exists in the groups with exponent \(39\). This fills two missing entries in Lopez and Sanchez’s table with answer “no”. Furthermore, if a Spence difference set \(D\) has Character Divisibility Property, then \(D\) is one of the difference sets constructed by Spence.

In this paper we concentrate on those graphs which are \((a, d)\)-face antimagic, and we show that the graphs \(D_n\) from a special class of convex polytopes consisting of \(4\)-sided faces are \((6n + 3, 2)\)-face antimagic and \((4n + 4, 4)\)-face antimagic. It is worth a conjecture, we feel, that \(D_n\) are \((2n + 5, 6)\)-face antimagic.

Let \(\{G(n,k)\}\) be a family of graphs where \(G(n, k)\) is the graph obtained from \(K_n\), the complete graph on \(n\) vertices, by removing any set of \(k\) parallel edges. In this paper, the lower bound for the multiplicity of triangles in any \(2\)-edge coloring of the family of graphs \(\{G(n, k)\}\) is calculated and it is proved that this lower bound is sharp when \(n \geq 2k + 4\) by explicit coloring schemes in a recursive manner. For the cases \(n = 2k + 1, 2k + 2\), and \(2k + 3\), this lower bound is not sharp and the exact bound in these cases are also independently calculated by explicit constructions.

In this paper we introduce a new parameter related to the index of convergence of Boolean matrices — the generalized index. The parameter is motivated by memoryless communication systems. We obtain the values of this parameter for reducible, irreducible and symmetric matrices.

In this paper we extend the definition of pseudograceful graphs given by Frucht [3] to all graphs \(G\) with vertex set \(V(G)\) and edge set \(E(G)\) such that
\(|V(G)| \leq |E(G)| + 1\) and we prove that if \(G\) is a pseudograceful graph, then \(G \cup K_{m,n}\).is pseudograceful
for \(m,n \geq 2\) and \((m,n) \neq (2,2)\) and is graceful for \(m,n \geq 2\). This enables us to obtain several new families of graceful and disconnected graphs.

A graph \(G\) is \(Z_m\)-well-covered if \(|I| \equiv |J| \pmod{m}\), for all \(I\), \(J\) maximal independent sets in \(V(G)\). A graph \(G\) is a \(1-Z_m\)-well-covered graph if \(G\) is \(Z_m\)-well-covered and \(G\setminus\{v\}\) is \(Z_m\)-well-covered, \(\forall v \in V(G)\). A graph \(G\) is strongly \(Z_m\)-well-covered if \(G\) is a \(Z_m\)-well-covered graph and \(G\setminus\{e\}\) is \(Z_m\)-well-covered, \(\forall e \in E(G)\). Here we prove some results about \(1-Z_m\)-well-covered and strongly \(Z_m\)-well-covered graphs.

There are two types of quadrangles in a projective plane, Fano quadrangles, and non-Fano quadrangles. The number of quadrangles in some small projective planes is counted according to type, and an interesting configuration in the Hughes plane is displayed.

Let \(S = T \sim (\cup\{A : A \in \mathcal{A}\})\), where \(T\) is a simply connected orthogonal polygon and \(\mathcal{A}\) is a collection of \(n\) pairwise disjoint open rectangular regions contained in \(T\). Point \(x\) belongs to the staircase kernel of \(S\), Ker \(S\), if and only if \(x\) belongs to Ker \(T\) and neither the horizontal nor the vertical line through \(x\) meets any \(A\) in \(\mathcal{A}\). This produces a Krasnosel’skii-type theorem for \(S\) in terms of \(n\). However, an example shows that, independent of \(n\), no general Krasnosel’skii number exists for \(S\).

We show that the secants of an arc of size near to \({\sqrt{2q}}\) cover almost half plane; also, a random union of \(log_2 q\) arcs of this size is such that its secants cover the plane.