Two \(G\)-designs \((X, \mathcal{A}_1)\) and \((X, \mathcal{A}_2)\) are said to intersect in \(m\) blocks if \(|\mathcal{A}_1 \cap \mathcal{A}_2| = m\). In this paper, we complete the solution of the intersection problem for \(G\)-designs, where \(G\) is a connected graph of size five which contains a cycle.

In this paper we discuss how the addition of a new edge affects the total edge irregularity strength of a graph.

Let \(G\) be a connected graph and \(k > 1\) be an integer. The local \(k\)-restricted edge connectivity \(\lambda_k(X,Y)\) of \(X,Y\) in \(G\) is the maximum number of edge-disjoint \(X\)-\(Y\) paths for \(X,Y \subseteq V\) with \(|X| = |Y| = k\), \(X \cap Y = \emptyset\), \(G[X]\) and \(G[Y]\) are connected. The \(k\)-restricted edge connectivity of \(G\) is defined as \(\lambda_k(G) = \min\{\lambda_k(X,Y) : X,Y \subseteq V, |X| = |Y| = k, X \cap Y = \emptyset, G[X] \text{ and } G[Y]\) are connected. Then \(G\) is local optimal \(k\)-restricted edge connected if \(\lambda_k(X,Y) = \min\{w(X), w(Y)\}\) for all \(X,Y \subseteq V\) with \(|X| = |Y| = k\), \(G[X]\) and \(G[Y]\) are connected, where \(w(X) = |E(X, \overline{X})|\). If \(\lambda_k(G) = \xi_k(G)\), where \(\xi_k(G) = \min\{w(X) : U \subset V, |U| = k \text{ and } G[U] \text{ is connected}\}\), then \(G\) is called \(\lambda_k\)-optimal. In this paper, we obtain several sufficient conditions for a graph to be \(3\)-optimal (or local optimal \(k\)-restricted edge connected).

A graph \(G\) is called edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \to \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv)\) is a constant for each \(uv \in E(G)\). Also, \(G\) is called super edge-magic if \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\). Moreover, the super edge-magic deficiency, \(\mu_s(G)\), of a graph \(G\) is defined to be the smallest nonnegative integer \(n\) with the property that the graph \(G \cup nK_1\) is super edge-magic, or \(+\infty\) if there exists no such integer \(n\). In this paper, we introduce the notion of the sequential number, \(\sigma(G)\), of a graph \(G\) without isolated vertices to be either the smallest positive integer \(n\) for which it is possible to label the vertices of \(G\) with distinct elements from the set \(\{0, 1, \ldots, n\}\) in such a way that each \(uv \in E(G)\) is labeled \(f(u) + f(v)\) and the resulting edge labels are \(|E(G)|\) consecutive integers, or \(+\infty\) if there exists no such integer \(n\). We prove that \(\sigma(G) = \mu_s(G) + |V(G)| – 1\) for any graph \(G\) without isolated vertices, and \(\sigma(K_{m,n}) = mn\) for every two positive integers \(m\) and \(n\), which allows us to settle the conjecture that \(\mu_s(K_{m,n}) = (m-1)(n-1)\) for every two positive integers \(m\) and \(n\).

Let \(G = (V, E)\) be a graph. An edge labeling \(f: E \to \mathbb{Z}_2\) induces a vertex labeling \(f^*: V \to \mathbb{Z}_2\) defined by \(f^*(v) = \sum_{uv \in E} f(uv) \pmod{2}\). For each \(i \in \mathbb{Z}_2\), define \(E_i(f) = |f^{-1}(i)|\) and \(V_i(f) = |(f^*)^{-1}(i)|\). We call \(f\) edge-friendly if \(|E_1(f) – E_0(f)| \leq 1\). The edge-friendly index \(I_f(G)\) is defined as \(V_1(f) – V_0(f)\), and the full edge-friendly index set \(FEFI(G)\) is defined as \(\{I_f(G): f \text{ is an edge-friendly labeling}\}\). Further, the edge-friendly index set \(EFI(G)\) is defined as \(\{|I_f(G)|: f \text{ is an edge-friendly labeling}\}\). In this paper, we study the full edge-friendly index set of the star \(K_{1,n}\), \(2\)-regular graph, wheel \(W_n\), and \(m\) copies of path \(mP_n\), \(m \geq 1\).

An acyclic total coloring is a proper total coloring of a graph \(G\) such that there are at least \(4\) colors on vertices and edges incident with a cycle of \(G\). The acyclic total chromatic number of \(G\), \(\chi”_a(G)\), is the least number of colors in an acyclic total coloring of \(G\). In this paper, we prove that for every plane graph \(G\) with maximum degree \(\Delta\) and girth \(g(G)\), \(\chi_a(G) = \Delta+1\) if (1) \(\Delta \geq 9\) and \(g(G) \geq 4\); (2) \(\Delta \geq 6\) and \(g(G) \geq 5\); (3) \(\Delta \geq 4\) and \(g(G) \geq 6\); (4) \(\Delta \geq 3\) and \(g(G) \geq 14\).

Codes in \(l_{p\gamma}\)-spaces, introduced by the author in [3], are a natural generalization of one-dimensional codes in \(RT\)-spaces [6] to block coding and have applications in different areas of combinatorial/discrete mathematics, e.g., in the theory of uniform distribution, experimental designs, cryptography, etc. In this paper, we introduce various types of weight enumerators in \(l_{p\gamma}\)-codes, viz., exact weight enumerator, complete weight enumerator, block weight enumerator, and \(\gamma\)-weight enumerator. We obtain the MacWilliams duality relation for the exact and complete weight enumerators of an \(l_{p\gamma}\)-code.

We introduce a theorem on bipartite graphs, and some theorems on chains of two and three complete graphs, considering when they are combination or non-combination graphs, present some families of combination graphs. We give a survey for trees of order \(\leq 10\), which are all combination graphs.

A set of vertices in a graph \(G\) without isolated vertices is a total dominating set (TDS) of \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\). The minimum cardinality of a TDS of \(G\) is the total domination number \(\gamma_t(G)\) of \(G\). In this paper, the total domination number of generalized \(n\)-graphs and \(m \times n\) ladder graphs is determined.

We identify a graph without proper cycles, which is comatching with a cycle,The result is then extended to certain general families of graphs with cyclomatic number \(1\), formed by attaching trees to cycles.