In this paper, we show that if \(G\) is an “\(\alpha\)-labeled” graph and if \(H\) is a “pseudograceful” graph, then \(G \cup H\) can be graceful or “pseudograceful” under some conditions on the \(\alpha\)-labeling function of \(G\). This generalizes Theorem 2.1 of [21]. We also show that if \(G\) is a Skolem-graceful, then \(G + \overline{K_n}\) is graceful for all \(n \geq 1\). We also give a partial answer to the question in [1] about the gracefulness of \(\overline{K_n} + mK_2\) for \(m \geq 3\). Finally, we complete the characterization of graceful graphs in the family \(C_m \cup S_n\).

We study the discrete version of the \(p\)-Laplacian operator — \(\textrm{div}(|\nabla u|^{p-2}\nabla u)\) — and we give some estimates of its smallest positive eigenvalue. In earlier papers, eigenvalues of the discrete Laplacian have been considered. We shall here study more general means. We shall also, in particular, study the case when the graph is complete. We give an estimate of the smallest positive eigenvalue of the \(p\)-Laplacian when the graph is a subgraph of \(\mathbb{Z}^n\) in this context. We give all eigenvalues of the \(p\)-Laplacian when the graph is complete.

Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as AT-free graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipartite permutation graphs by showing that any connected graph in the class can be stretched into a “path” with “edges” being chain graphs. A particular consequence from the obtained characterization is that the clique-width of bipartite permutation graphs is unbounded, which refines a recent result of Golumbic and Rotics for permutation graphs.

A known result due to Matthews and Sunner is that every \(2\)-connected claw-free graph on \(n\) vertices contains a cycle of length at least \(\min\{2\delta+4,n\}\), and is Hamiltonian if \(n \leq 3\delta+2\). In this paper, we show that every \(2\)-connected claw-free graph on \(n\) vertices which does not belong to one of three classes of exceptional graphs contains a cycle of length at least \(\min\{4\delta-2,n\}\), hereby generalizing several known results. Moreover, the bound \(4\delta-2\) is almost best possible.

A graph or a digraph \(G\) is called super-edge-connected or super-\(\lambda\), if every minimum edge cut consists of edges adjacent to or from a vertex of minimum degree. Clearly, if \(G\) is super-\(\lambda\), then \(\lambda(G) = \delta(G)\), where \(\delta(G)\) is the minimum degree and \(\lambda(G)\) is the edge-connectivity of \(G\).

In this paper, degree sequence conditions for graphs and digraphs as well as for bipartite graphs and digraphs to be super-\(\lambda\) are presented.

Given integers \(k \geq 2\) and \(n \geq k\), let \(e(n, k)\) denote the maximum possible number of edges in an \(m\)-vertex graph which has no \(k\)-connected subgraph. It is immediate that \(e(n, 2) = n – 1\). Mader [2] conjectured that for every \(k \leq 2\), if \(n\) is sufficiently large then \(c(n, k) \leq (1.5k-2)(n – k + 1),\) where equality holds whenever \(k – 1\) divides \(n\). In this note we prove that when \(n\) is sufficiently large then \(e(n, k) \leq \frac{193}{120}(k – 1)(n – k + 1) < 1.61(k – 1)(n – k + 1),\) thereby coming rather close to the conjectured bound.

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.