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.

A vertex \(v \in V(G)\) is said to be a self vertex switching of \(G\) if \(G\) is isomorphic to \(G^v\), where \(G^v\) is the graph obtained from \(G\) by deleting all edges of \(G\) incident to \(v\) and adding all edges incident to \(v\) which are not in \(G\). In [6], the author characterized connected unicyclic graphs each with a self vertex switching. In this paper, we characterize disconnected unicyclic graphs each with a self vertex switching.

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. A multi-wheel graph is a graph obtained from \(s\) cycles \(C_{n_1}, C_{n_2}, \ldots, C_{n_s}\) (\(s \geq 1\)) by adding a new vertex, say \(w\), and edges joining \(w\) to all the vertices of the \(s\) cycles. In this article, we solve a conjecture posed by Yu et al. in 2006 and prove that it is not always true. Furthermore, the classification problem of multi-wheel graphs on \(f\)-colorings is solved completely.

For a connected graph \(G = (V, E)\) of order at least two, a chord of a path \(P\) is an edge joining two non-adjacent vertices of \(P\). A path \(P\) is called a monophonic path if it is a chordless path. A longest \(x\)-\(y\) monophonic path is called an \(x\)-\(y\) detour monophonic path. A set \(S\) of vertices of \(G\) is a detour monophonic set of \(G\) if each vertex \(v\) of \(G\) lies on an \(x\)-\(y\) detour monophonic path for some \(x\) and \(y\) in \(S\). The minimum cardinality of a detour monophonic set of \(G\) is the detour monophonic number of \(G\) and is denoted by \(dm(G)\). For any two vertices \(u\) and \(v\) in \(G\), the monophonic distance \(dm(u,v)\) from \(u\) to \(v\) is defined as the length of a \(u\)-\(v\) detour monophonic path in \(G\). The monophonic eccentricity \(em(v)\) of a vertex \(v\) in \(G\) is the maximum monophonic distance from \(v\) to a vertex of \(G\). The monophonic radius \(rad_{m}(G)\) of \(G\) is the minimum monophonic eccentricity among the vertices of \(G\), while the monophonic diameter \(diam_{m}(G)\) of \(G\) is the maximum monophonic eccentricity among the vertices of \(G\). It is shown that for positive integers \(r\), \(d\), and \(n \geq 4\) with \(r < d\), there exists a connected graph \(G\) with \(rad_{m}(G) = r\), \(diam_{m}(G) = d\), and \(dm(G) = n\). Also, if \(p\), \(d\), and \(n\) are integers with \(2 \leq n \leq p-d+4\) and \(d \geq 3\), there is a connected graph \(G\) of order \(p\), monophonic diameter \(d\), and detour monophonic number \(n\). Further, we study how the detour monophonic number of a graph is affected by adding some pendant edges to the graph.