In this paper, we give a few applications of combinatorial design theory to a few problems in extremal graph theory. Using known results in combinatorial design theory, we have unified, simplified, and extended results on a few problems.

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(\overline{f}: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(f(uv) = |f(u) – f(v)|\). Let \(v_f(0),v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0),e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – vf(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).

A \(t\)-uniform homeomorph \(P_t(G)\) of \(G\) is the graph obtained by replacing all edges of \(G\) by vertex disjoint paths of length \(t\). In this paper we show that (1)\(P_t(K_{2n})\) is cordial for all \(t \geq 2\).(2) \(P_t(K_{2n+1})\) is cordial if and only iff (a) \(t \equiv 0 \pmod{4}\), or(b) \(t\) is odd and \(n\) is not \(\equiv 2 \pmod{4}\), or (c) \(t \equiv 2 \pmod{4}\) and \(n\) is even.

For any positive integer \(k\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_k\)-magic if there exists a labeling \(l: E(G) \to \mathbb{Z}_k – \{0\}\) such that the induced vertex set labeling \(l^+: V(G) \to \mathbb{Z}_k\) defined by

\[l^+(v) = \sum\{l(uv): uv \in E(G)\}\]

is a constant map. For a given graph \(G\), the set of all \(h \in \mathbb{Z_+}\) for which \(G\) is \(\mathbb{Z}_h\)-magic is called the integer-magic spectrum of \(G\) and is denoted by \(IM(G)\). In this paper, we will determine the integer-magic spectra of the graphs which are formed by the amalgamation of stars and cycles. In particular, we will provide examples of graphs that for a given \(n > 2\), they are not \(h\)-magic for all values of \(2 \leq k \leq n\).

In this paper, various transformations of the set of closed meanders are introduced. Some of these are used in order to partition the above set and to find a representative of each class. Furthermore, each closed meander is separated into shorter ones.

We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a crease pattern if it represents the crease lines needed to fold a piece of paper into something. A flat fold is a crease pattern which lies flat when folded, i.e., can be pressed in a book without crumpling. Given a crease pattern \(C = (V, E)\), a mountain-valley (MV) assignment is a function \(f : E \to \{M, V\}\) which indicates which crease lines are convex and which are concave, respectively. A MV assignment is valid if it doesn’t force the paper to self-intersect when folded. We examine the problem of counting the number of valid MV assignments for a given crease pattern. In particular, we develop recursive functions that count the number of valid MV assignments for flat vertex folds, crease patterns with only one vertex in the interior of the paper. We also provide examples, especially those of Justin, that illustrate the difficulty of the general multivertex case.

An asteroidal triple is an independent set of three vertices in a graph such that every two of them are joined by a path avoiding the closed neighborhood of the third. Graphs without asteroidal triples are called AT-free graphs. In this paper, we show that every AT-free graph admits a vertex ordering that we call a \(2\)-cocomparability ordering. The new suggested ordering generalizes the cocomparability ordering achievable for cocomparability graphs. According to the property of this ordering, we show that every proper power \(G^k\) (\(k \geq 2\)) of an AT-free graph \(G\) is a cocomparability graph. Moreover, we demonstrate that our results can be exploited for algorithmic purposes on AT-free graphs.

We exhibit some problems definable in Feder and Vardi’s logic \(MMSNP\) that are not in the class \(CSP\) of constraint satisfaction problems. Whilst some of these problems have previously been shown to be in \(MMSNP\) (that is, definable in \(MMSNP\)) but not in \(CSP\), existing proofs are probabilistic in nature. We provide explicit combinatorial constructions to prove that these problems are not in \(CSP\) and we use these constructions to exhibit yet more problems in \(MMSNP\) that are not in \(CSP\).

The distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining \(u\) and \(v\). The eccentricity \(e(v)\) of vertex \(v\) is the distance to a vertex farthest from \(v\). In a graph \(G\), an eccentric vertex of \(v\) is a vertex farthest from \(v\), that is, a vertex \(u\) for which \(d(u,v) = e(v)\). Given a set \(X\) of vertices in \(G\), the vertices of \(X\) are mutually eccentric provided that for any pair of vertices \(u\) and \(v\) in \(X\), \(u\) is an eccentric vertex of \(v\) and \(v\) is an eccentric vertex of \(u\). In this paper, we discuss problems concerning sets of mutually eccentric vertices in graphs.

A \(k\)-circular-distance-two labeling (or \(k\)-c-labeling) of a simple graph \(G\) is a vertex-labeling, using the labels \(0, 1, 2, \ldots, k-1\), such that the “circular difference” (mod \(k\)) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The \(\sigma\)-number, \(\sigma(G)\), of a graph \(G\) is the minimum \(k\) of a \(k\)-c-labeling of \(G\). For any given positive integers \(n\) and \(k\), let \(\mathcal {G}^{\sigma}(n, k)\) denote the set of graphs \(G\) on \(n\) vertices and \(\sigma(G) = k\). We determine the maximum size (number of edges) and the minimum size of a graph \(G \in \mathcal {G}^{\sigma}(n, k)\). Furthermore, we prove that for any value \(p\) between the maximum and the minimum size, there exists a graph \(G \in \mathcal {G}^{\sigma}(n, k)\) of size \(p\). These results are analogues of the ones by Georges and Mauro [4] on distance-two labelings.

We give a parametric representation for generic magic squares. This makes it relatively easy to construct magic squares having desired properties. It also suggests a convenient method for generating and classifying all the magic squares of every given order.