A graph \(G\) with vertex set \(V(G)\) is an exact \(n\)-step domination graph if there is some subset \(S \subseteq V(G)\) such that each vertex in \(G\) is distance \(t\) from exactly one vertex in \(S\). Given a set \(A \subseteq \mathbb{N}\), we characterize cycles \(C_t\) with sets \(S \subseteq V(C_t)\) that are simultaneously \(a\)-step dominating for precisely those \(a \in A\). Using Polya’s method, we compute the number of \(t\)-step dominating sets for a cycle \(C_t\) that are distinct up to automorphisms of \(C_t\). Finally, we generalize the notion of exact \(t\)-step domination.

Let \(D\) be a digraph. The competition-common enemy graph of \(D\) has the same set of vertices as \(D\) and an edge between vertices \(u\) and \(v\) if and only if there are vertices \(w\) and \(x\) in \(D\) such that \((w,u), (w,v), (u,x)\), and \((v,x)\) are arcs of \(D\). We call a graph a CCE-graph if it is the competition-common enemy graph of some digraph. We also call a graph \(G = (V, E)\) CCE-orientable if we can give an orientation \(F\) of \(G\) so that whenever \((w,u), (w,v), (u,x)\), and \((v,x)\) are in \(F\), either \((u,v)\) or \((v,u)\) is in \(F\). Bak \(et\; al. [1997]\) found a large class of graphs that are CCE-orientable and proposed an open question of finding graphs that are not CCE-orientable. In this paper, we answer their question by presenting two families of graphs that are not CCE-orientable. We also give a CCE-graph that is not CCE-orientable, which answers another question proposed by Bak \(et \;al. [1997]\). Finally, we find a new family of graphs that are CCE-orientable.

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.

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.