Hrnciar and Haviar [3] gave a method to construct a graceful labeling for all trees of diameter at most five. Based on their method and the methods described in Balbuena et al. [1], we shall describe a new construction for gracefully labeled trees by attaching trees to the vertices of a tree admitting a bipartite graceful labeling.

Given two graphs \( G \) and \( H \) and a function \( f \subset V(G) \times V(H) \), Hedetniemi [9] defined the \emph{function graph} \( GfH \) by \( V(GfH) = V(G) \cup V(H) \) and \( E(GfH) = E(G) \cup E(H) \cup \{uv \mid v : f(u)\} \). Whenever \( G \cong H \), the function graph was called a functigraph by Chen, Ferrero, Gera, and Yi [7]. A function graph is a generalization of the \( \alpha \)-permutation graph introduced by Chartrand and Harary [5]. The independence number of a graph is the size of a largest set of mutually non-adjacent vertices. In this paper, we study independence number in function graphs. In particular, we give a lower bound in terms of the order and the chromatic number, which improves on some elementary results and has a number of interesting corollaries.

A \( k \)-edge labeling of a graph \( G \) is a function \( f \) from the edge set \( E(G) \) to the set of integers \(\{0, \ldots, k-1\}\). Such a labeling induces a labeling \( f \) on the vertex set \( V(G) \) by defining \( f(v) = \sum f(e) \), where the summation is taken over all the edges incident on the vertex \( v \) and the value is reduced modulo \( k \). Cahit calls this labeling edge-\( k \)-equitable if \( f \) assigns the labels \(\{0, \ldots, k-1\}\) equitably to the vertices as well as edges.

If \( G_1, \ldots, G_T \) is a family of graphs each having a graph \( H \) as an induced subgraph, then by \( H \)-union \( G \) of this family we mean the graph obtained by identifying all the corresponding vertices as well as edges of the copies of \(H\) in \(G_1, \ldots, G_T\).

In this paper, which is a sequel to the paper entitled `On edge-\(3\)-equitability of \(\overline{K}_n\)-union of gears’, we prove that \(\overline{K}_n\)-union of copies of helm \(H_n\) is edge-\(3\)-equitable for all \(n \geq 6\).

A graceful labeling of a graph \( G \) with \( q \) edges is an injective assignment from the vertices of \( G \) into \(\{0, 1, \ldots, q\}\) such that when each edge is assigned the absolute value of the difference of the vertex labels it connects, the resulting edge labels are distinct. In 1978, Frucht conjectured that for gracefully labeled coronas \( C_n \odot K_1 \), the omitted vertex label is always even. In this paper, we will verify Frucht’s conjecture.

Let \( \mathcal{C} = \{C_1, \ldots, C_n\} \) be a family of distinct boxes in \( \mathbb{R}^d \), and let \( S = C_1 \cup \cdots \cup C_n \). Assume that \( S \) is staircase starshaped. If the intersection graph of \( \mathcal{C} \) is a tree, then the staircase kernel of \( S \), \( \ker S \), will be staircase convex. However, an example in \( \mathbb{R}^3 \) reveals that, without this requirement on the intersection graph of \( \mathcal{C} \), components of \( \ker S \) need not be staircase convex. Thus the structure of the kernel in higher dimensional staircase starshaped sets provides a striking contrast to its structure in planar sets.

We study cube-complementary graphs, that is, graphs whose com- plement and cube are isomorphic. We prove several necessary conditions for a graph to be cube-complementary, describe ways of building new cube-complementary graphs from existing ones, and construct
infinite families of cube-complementary graphs.

We suggest the notion of the surface area centered at an edge for a network structure, which generalizes the usual notion of surface area of a structure centered at a vertex. As a specific result, we derive explicit expressions of the edge-centered surface areas for the edge-asymmetric \((n, k)\)-star graph, following a generating function approach, in terms of two different kinds of edges.

For a set \( S \) of \( k \) vertices of \( G \), let \( \kappa(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( V(T_i) \cap V(T_j) = S \) for \( 1 \leq i \neq j \leq \ell \) and \( \lambda(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( S \subseteq V(T_i) \) for \( 1 \leq i \leq \ell \). Similar to the classical maximum local connectivity, H. Li et al. introduced the parameter \( \overline{\kappa}_k(G) = \max\{\kappa(S) \mid S \subseteq V(G), |S| = k\} \), which is called the maximum generalized local connectivity of \( G \). The maximum generalized local edge-connectivity of \( G \) which was introduced by X. Li et al. is defined as \( \overline{\lambda}_k(G) = \max\{\lambda(S) \mid S \subseteq V(G), |S| = k\} \). In this paper, we investigate the maximum generalized local connectivity and edge-connectivity of a cubic connected Cayley graph \( G \) on an Abelian group. We determine the precise values for \( \overline{\kappa}_3(G) \) and \( \overline{\lambda}_3(G) \) and also prove

The generalized \( k \)-connectivity \( \kappa_k(G) \) of a graph \( G \) was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized \( k \)-edge-connectivity. In this paper, we completely determine the precise values of the generalized \( 3 \)-connectivity and generalized \( 3 \)-edge-connectivity for the Cartesian products of some graph classes.

In this work, we investigate the gap-adjacent-chromatic number, a graph colouring parameter introduced by M. A. Tahraoui, E. Duchéne, and H. Kheddouci in \([5]\). From an edge labelling \( f: E \to \{1, \ldots, k\} \) of a graph \( G = (V, E) \), the vertices of \( G \) get an induced colouring. Vertices of degree greater than one are coloured with the difference between their maximum and their minimum incident edge label, i.e., with their so-called gap, and vertices of degree one get their incident edge label as colour. The gap-adjacent-chromatic number of \( G \) is the minimum \( k \) for which a labelling \( f \) of \( G \) exists that induces a proper vertex colouring.

The main purpose of this work is to state easy colouring approaches for bipartite graphs and to estimate the gap-adjacent-chromatic number for arbitrary graphs in terms of the chromatic number.