In this paper, we show the necessary and sufficient conditions for a complete graph on \(n\) vertices with a hole of size \(v\) (\(K_n \setminus K_v\)) to be decomposed into isomorphic copies of \(K_3\) with a pendant edge.

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.