Fishburn, Tanenbaum, and Trenk define the linear discrepancy \(\text{Id}(P)\) of a poset \( P = (V, <_P) \) as the minimum integer \( k \geq 0 \) for which there exists a bijection \( f : V \rightarrow \{1,2,\ldots,|V|\} \) such that \( u <_P v \) implies \( f(u) < f(v) \) and \( u ||_P v \) implies \( |f(u) – f(v)| \leq k \). In their work, they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph. Here we provide partial solutions to some problems formulated in their study about the linear discrepancy and the bandwidth of cocomparability graphs.

To date, investigations on critical sets for a set of mutually orthogonal Latin squares (MOLS) have been carried out only for small orders \( \leq 9 \). In this paper, we deal with a pair of cyclic orthogonal Latin squares of order \( n \), \( n \geq 11 \), \( n \) odd. Through the construction of a uniquely completable set, we give an upper bound on the size of the minimal critical set. In particular, for \( n = 15 \), a critical set achieving this bound is obtained.

We improve the lower bound for the \(\alpha\)-size of trees with maximum degree three.

A graph \( G \) is called \((a,d)\)-\(\text{edge antimagic total}\) \((a, d)\)-\(\text{EAT}\) if there exist integers \( a > 0, d \geq 0 \) and a bijection \( \lambda: V \cup E \rightarrow \{1,2,\ldots,|V| + |E|\} \) such that \(W = \{w(xy) : xy \in E\} = \{a,a+d,\ldots,a+(|E|-1)d\},\) where \( w(xy) = \lambda(x) + \lambda(y) + \lambda(xy) \). An \((a, d)\)-EAT labeling \( \lambda \) of graph \( G \) is \text{super} if \( \lambda(V) = \{1,2,\ldots,|V|\} \). In this paper, we describe how to construct a super \((a, d)\)-EAT labeling on some classes of disconnected graphs, namely \( P_n \cup P_{n+1} \), \( nP_2 \cup P_n \), and \( nP_2 \cup P_{n+2} \), for positive integer \( n \).

A graph \( G(V, E) \) is called a sum graph if there is an injective labeling called sum labeling \( L \) from \( V \) to a set of distinct positive integers \( S \) such that \( xy \in E \) if and only if there is a vertex \( w \in V \) such that \( L(w) = L(x) + L(y) \in S \). In such a case, \( w \) is called a working vertex. Every graph can be made into a sum graph by adding some isolated vertices, if necessary. The smallest number of isolated vertices that need to be added to a graph \( H \) to obtain a sum graph is called the sum number of \( H \); it is denoted by \( \sigma(H) \). A sum labeling which realizes \( H \cup \overline{K_\sigma(G)} \) as a sum graph is called an optimal sum labeling of \( H \).

Sum graph labeling offers a new method for defining graphs and for storing them digitally. Traditionally, a graph is defined as a set of vertices and a set of edges, specified by pairs of vertices which are the endpoints of an edge. To record a graph on a computer, the edges are usually stored either in the form of an adjacency matrix or as a linked list. Using sum graph labeling, we only need to store the set of vertices, together with some additional isolates, if needed. While previously the edges in a graph were specified explicitly, using sum graphs, edges can be specified implicitly.

A sum labeling \( L \) is called an exclusive sum labeling with respect to a subgraph \( H \) of \( G \) if \( L \) is a sum labeling of \( G \) where \( H \) contains no working vertex. The exclusive sum number \( \epsilon(H) \) of a graph \( H \) is the smallest number \( r \) such that there exists an exclusive sum labeling \( L \) which realizes \( H \cup \overline{K_{r}} \) as a sum graph. A labeling \( L \) is an optimal exclusive sum labeling of a graph \( H \) if \( L \) is a sum labeling of \( H \cup K_{\epsilon(H)} \) and \( H \) contains no working vertex. While the exclusive sum number is never smaller than the corresponding sum number of a graph, labeling graphs exclusively has other desirable features which give greater scope for combining two or more labeled graphs.

In this paper, we introduce exclusive sum graph labeling and we construct optimal exclusive sum graph labeling for complete bipartite graphs, paths, and cycles. The paper concludes with a summary of known results in exclusive sum labeling and exclusive sum numbers for several classes of graphs.

A total vertex irregular labeling of a graph \( G \) with \( v \) vertices and \( e \) edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of \( G \), denoted by \( tvs(G) \), is the minimum value of the largest label over all such irregular assignments. In this paper, we consider the total vertex irregular labeling of complete bipartite graphs \( K_{m,n} \) and prove that

\[
tvs(K_{m,n}) \geq \max \left\{ \left\lceil \frac{m+n}{m+1} \right\rceil, \left\lceil \frac{2m+n-1}{n} \right\rceil \right\} \quad \text{if } (m,n) \neq (2,2).
\]

For given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the smallest natural number \( n \) such that for every graph \( F \) of order \( n \): either \( F \) contains \( G \) or the complement of \( F \) contains \( H \).

This paper investigates the Ramsey number \( R(S_n, W_m) \) of stars versus wheels. We show that if \( m \) is odd, \( n \geq 3 \), and \( m \leq 2n-1 \), then \( R(S_n, W_m) = 3n-2 \). Furthermore, if \( n \) is odd and \( n \geq 5 \), then \( R(S_n, W_m) = 3n – \mu \), where \( \mu = 4 \) if \( m = 2n – 4 \) and \( \mu = 6 \) if \( m = 2n – 8 \) or \( m = 2n – 6 \).