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 some results of \(\overline{\kappa}_k(G)\) and \(\overline{\lambda}_k(G)\) for general \(k\).

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.

We call \( T = (G_1, G_2, G_3) \) a graph-triple of order \( t \) if the \( G_i \) are pairwise non-isomorphic graphs on \( t \) non-isolated vertices whose edges can be combined to form \( K_t \). If \( m \geq t \), we say \( T \) divides \( K_m \) if \( E(K_m) \) can be partitioned into copies of the graphs in \( T \) with each \( G_i \) used at least once, and we call such a partition a \( T \)-multidecomposition. For each graph-triple \( T \) of order \( 6 \) for which it was not previously known, we determine all \( K_m \), \( m \geq 6 \), that admit a \( T \)-multidecomposition. Moreover, we determine maximum multipackings and minimum multicoverings when \( K_m \) does not admit a multidecomposition.

For the Firefighter Process with weights on the vertices, we show that the problem of deciding whether a subset of vertices of a total weight can be saved from burning remains NP-complete when restricted to binary trees. In addition, we show that a greedy algorithm that defends the vertex of highest degree adjacent to a burning vertex is not an \(\epsilon\)-\emph{approximation} algorithm for any \(\epsilon \in (0, 1]\) for the problem of determining the maximum weight that can be saved. This closes an open problem posed by MacGillivray and Wang.