In this paper, we study the properties of super edge-magic total graphs. In particular, we propose some algorithms to construct new super edge-magic total graphs from the old ones. We also construct a super edge-magic total labeling on certain disconnected graphs, namely \( P_n \cup P_{n+1} \), \( nP_2 \cup P_n \), and \( nP_2 \cup P_{n+2} \).

Let \( G = G(v, e) \) be a finite simple graph with \( v \) vertices and \( e \) edges. An \((a, d)\)-\text{edge-antimagic-vertex} (EAV) \text{labeling} is a one-to-one mapping \( f: V(G) \to \{1, 2, \ldots, v\} \) such that for every edge \( xy \in E(G) \), the edge-weight set

\[
\{f(x) + f(y) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}
\]

for some positive integers \( a \) and \( d \). An \((a, d)\)-\text{edge-antimagic-total labeling} is a one-to-one mapping \( f: V(G) \cup E(G) \to \{1, 2, \ldots, v+e\} \) with the property that for every edge \( xy \in E(G) \),

\[
\{f(x) + f(y) + f(xy) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}.
\]

This labeling is called \text{super \((a, d)\)-edge-antimagic total labeling} if \( f(V(G)) = \{1, 2, \ldots, v\} \). In this paper, we investigate the relationship between the adjacency matrix, \((a, d)\)-edge-antimagic vertex labeling, and super \((a, d)\)-edge-antimagic total labeling, and show how to manipulate this matrix to construct new \((a, d)\)-edge-antimagic vertex labelings and new super \((a, d)\)-edge-antimagic total graphs.

A total labeling of a graph \( G \) with \( p \) vertices and \( q \) edges is a one-to-one mapping from \( V(G) \cup E(G) \) onto \( \{1,2,\ldots,p+q\} \). If the edge-weights (resp. vertex-weights) form an arithmetic progression starting from \( a \) and having common difference \( d \), then the labeling is called an \( (a,d) \)-edge (resp. vertex) – antimagic total labeling. In this paper, we consider such labeling applied to the generalized Petersen graph.

A simple graph \( G = (V,E) \) admits an \( H \)-\emph{covering} if every edge in \( E \) belongs to a subgraph of \( G \) isomorphic to \( H \). In this case, we say that \( G \) is \( H \)-\emph{magic} if there is a total labeling \( f : V \cup E \rightarrow \{1,2,\ldots,|V|+|E|\} \) such that for each subgraph \( H’ = (V’,E’) \) of \( G \) isomorphic to \( H \),

\[
\sum_{v \in V’} f(v) + \sum_{e \in E’} f(e)
\]

is constant. When \( f(V) = \{1,\ldots,|V|\} \), we say that \( G \) is \( H \)-\emph{supermagic}.

We study \( H \)-magic graphs for several classes of connected graphs. We also provide constructions of infinite families of \( H \)-magic graphs for an arbitrary given graph \( H \).

Let \( \lambda \) be an edge-magic total (EMT) labeling of graph \( G(V, E) \). Let \( W \subset V(G) \cup E(G) \). Any restriction of \( \lambda \) to \( W \) is called a \emph{partial EMT labeling} on \( G \). A partial EMT labeling \( \pi \) is a critical set in \( \lambda \) if \( \lambda \) is the only edge-magic total labeling having \( \pi \) as its partial EMT labeling, and no proper restriction of \( \pi \) satisfies the first condition. In this paper, we study the property of critical sets in such a labeling. We determine critical sets in an EMT labeling for a given graph \( G \).

A \( (p,q) \) graph \( G \) is called edge-magic if there exists a bijective function \( f : V(G) \cup E(G) \rightarrow \{1,2,\ldots,p+q\} \) such that \( f(u) + f(v) + f(uv) \) is a constant for each edge \( uv \in E(G) \). Also, \( G \) is said to be super edge-magic if \( f(V(G)) = \{1,2,\ldots, p\} \). Furthermore, the super edge-magic deficiency, \( \mu_s(G) \), of a graph \( G \) is defined to be either 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, the super edge-magic deficiency of certain forests and 2-regular graphs is computed, which in turn leads to some conjectures on the super edge-magic deficiencies of graphs in these classes. Additionally, some edge-magic deficiency analogues to the super edge-magic deficiency results on forests are presented.

Suppose \( G = (V,E,F) \) is a finite plane graph with vertex set \( V(G) \), edge set \( E(G) \), and face set \( F(G) \). A bijection \( \lambda: V(G) \cup E(G) \cup F(G) \rightarrow \{1,2,3,\ldots,|V(G)| + |E(G)| + |F(G)|\} \) is called a labeling of type \( (1,1,1) \). The weight of a face under a labeling is the sum of the labels (if present) carried by that face and the edges and vertices surrounding it. A labeling of a plane graph \( G \) is called \( d \)-\emph{antimagic} if for every number \( s \geq 3 \), the set of \( s \)-sided face weights is

\[
W_s = \{a_s + id: 0 \leq i \leq f_s\}
\]

for some integers \( a_s \) and \( d \) (\( a > 0 \), \( d \geq 0 \)), where \( f_s \) is the number of \( s \)-sided faces. We allow different sets \( W_s \) for different \( s \).

In this paper, we deal with \( d \)-\emph{antimagic} labelings of type \( (1,1,1) \) for a special class of plane graphs \( C_a^b \) and we show that a \( C_a^b \) graph has \( d \)-antimagic labeling for \( d \in \{a-2,a-1,a+1,a+2\} \).

A \( V(m,t) \) leads to \( m \) idempotent pairwise orthogonal Latin squares of order \( (m+1)t+1 \) with one common hole of order \( t \). \( V(m,t) \)’s can also be used to construct perfect Mendelsohn designs and optimal optical orthogonal codes. For \( 3 \leq m \leq 8 \), the spectrum for \( V(m,t) \) has been determined. In this article, we investigate the existence of \( V(m,t) \) with \( m = 9 \) and show that a \( V(9,t) \) always exists in \( GF(q) \) for any prime power \( q = 9t + 1 \) with the exception of \( q = 73 \) and one possible exception of \( q = 5^6 \).

A Vertex Magic Total Labeling of a graph \( G \) is a one-to-one map \( \lambda \) from \( E(G) \cup V(G) \) onto the set of integers \( \{1, 2, \ldots, e + v\} \) such that for all \( x \in V \) we have

\[
\lambda(x) + \sum \lambda(xy) = h
\]

for some constant \( h \), where the sum is taken over all vertices \( y \) adjacent to \( x \). In this paper, we present several theorems on the existence of such labelings for multipartite graphs and give constructions for labelings for two infinite families of complete tripartite graphs, namely \( K_{1,n,n} \) for odd \( n \) and \( K_{2,n,n} \) for \( n \equiv 3 \pmod{4} \).