Let a set \([n] = \{1,2,\ldots,n\}\) be given. Finding a subset \( S \) of \( 2^{[n]} \) with minimum cardinality such that, for any two distinct elements \( x, y \in [n] \), there exist disjoint subsets \( A_x, A_y \in \mathcal{S} \) such that \( x \in A_x \) and \( y \in A_y \) is called the \emph{extremal set} problem. In this paper, we define the Extremal Set Decision (ESD) Problem and study its complexity.

A cyclic base ordering of a connected graph \( G \) is a cyclic ordering of \( E(G) \) such that every \( |V(G)| – 1 \) cyclically consecutive edges form a spanning tree of \( G \). Let \( G \) be a graph with \( E(G) \neq \emptyset \) and let \( \omega(G) \) denote the number of components in \( G \). The invariants \( d(G) \) and \( \gamma(G) \) are respectively defined as \( d(G) = \frac{|E(G)|}{|V(G)| – \omega(G)} \) and \( \gamma(G) = \text{max}\{d(H)\} \), where \( H \) runs over all subgraphs of \( G \) with \( E(H) \neq \emptyset \). A graph \( G \) is uniformly dense if \( d(G) = \gamma(G) \). Kajitani et al. [8] conjectured in 1988 that a connected graph \( G \) has a cyclic base ordering if and only if \( G \) is uniformly dense. In this paper, we show that this conjecture holds for some classes of uniformly dense graphs.

A graph \( G \) is a \((t, r)\)-regular graph if every collection of \( t \) independent vertices is collectively adjacent to exactly \( r \) vertices. Let \( p, s \), and \( m \) be positive integers, where \( m \geq 2 \), and let \( G \) be a \((2, r)\)-regular graph. If \( n \) is sufficiently large, then \( G \) is isomorphic to \( K_s + mK_p \), where \( 2(p-1) + s = r \). A nested \((2, r)\)-regular graph is constructed by replacing selected cliques in a \((2, r)\)-regular graph with a \((2, r’)\)-regular graph and joining the vertices of the peripheral cliques. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.

Given a graph \( G \), we show how to compute the number of (perfect) matchings in the graphs \( G \Box P_n \) and \( G \Box C_n \), by looking at appropriate entries in a power of a particular matrix. We give some generalizations and extensions of this result, including showing how to compute tilings of \( k \times n \) boards using monomers, dimers, and \( 2 \times 2 \) tiles.

Consider a simple undirected graph \( G = (V, E) \). A family of subtrees, \(\{T_v\}_{v \in V}\), of a tree \(\mathcal{T}\) is called a \((\mathcal{T}; t)\)-representation of \(G\) provided \( uv \in E \) if and only if \( |T_u \cap T_v| \geq t \). In this paper, we consider \((\mathcal{T}; t)\)-representations for graphs containing large asteroidal sets, where \(\mathcal{T}\) is a subdivision of the \(n\)-star \(K_{1, n}\). An asteroidal set in a graph \(G\) is a subset \(A\) of the vertex set such that for all 3-element subsets of \(A\), there exists a path in \(G\) between any two of these vertices which avoids the neighborhood of the third vertex. We construct a representation of an asteroidal set of size \( n + \sum_{k=2}^{n} \binom{n}{k} \binom{t-2}{k-1} \) and show that no graph containing a larger asteroidal set can be represented.

A set \( S \) of vertices in a graph \( G \) is a total dominating set of \( G \) if every vertex of \( G \) is adjacent to some vertex in \( S \). The minimum cardinality of a total dominating set of \( G \) is the total domination number of \( G \). We study graphs having the same total domination number as their complements. In particular, we characterize the cubic graphs having this property. Also, we characterize such graphs with total domination numbers equal to two or three, and we determine properties of the ones with larger total domination numbers.

In 1991 Gnanajothi conjectured: Each tree is odd-graceful. In this paper, we define the edge-ordered odd-graceful labeling of trees and show the odd-gracefulness of all symmetric trees.

A method is suggested for the construction of quadrangulations of the closed orientable surface with given genus \( g \) and either (1) with a given chromatic number or (2) with a given order allowed by the genus \( g \). In particular, N. Hartshfield and G. Ringel’s results [J. Comb. Theory, Ser. B 46 (1989), 84-95] are generalized by way of generating minimal quadrangulations of infinitely many other genera.

For integers \( s, t \geq 1 \), the Ramsey number \( R(s, t) \) is defined to be the least positive integer \( n \) such that every graph on \( n \) vertices contains either a clique of order \( s \) or an independent set of order \( t \). In this note, the lower bound for the Ramsey number \( R(7, 9) \) is improved from \( 241 \) to \( 242 \). The new bound is obtained by searching the maximum common induced subgraph between two graphs with a depth variable local search technique.

In this paper, we give an alternative and more intuitive proof to one of two classic inequalities given by Diaconis and Graham in 1977. The inequality involves three metrics on the symmetric group, i.e., the set of all permutations of the first \( n \) positive integers. Our technique for the proof of the inequality allows us to resolve an open problem posed in that paper: When does equality hold? It also allows us to estimate how often equality holds. In addition, our technique can sometimes be applied for the proof of other inequalities between metrics or pseudo-metrics on the symmetric group.