We introduce a generalization of the well-known concept of graceful labeling. Given a graph \(\Gamma\) with \(e = d.m\) edges, we define a \(d\)-graceful labeling of \(G\) as an injective function \(f: V(G) \rightarrow \{0, 1, 2, \ldots, d(m+1) – 1\}\) such that \(\{|f(x) – f(y)| : \{x, y\} \in E(\Gamma)\}\) = \(\{1, 2, 3, \ldots, d(m+1) – 1\} – \{m+1, 2(m+1), \ldots, (d-1)(m+1)\}.\) In the case of \(d = 1\) and of \(d = e\) we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively.Also, we call \(d\)-graceful \(\alpha\)-labeling of a bipartite graph \(\Gamma\) a \(d\)-graceful labeling of \(\Gamma\) with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other
one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of \(d\)-graceful \(\alpha\)-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.

Given a graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph.In this paper, we characterize potentially \(K_6 – E(K_3)\)-graphic sequences without zero terms, where \(K_6 – E(K_3)\) denotes the graph obtained from a complete graph on \(6\) vertices by deleting three edges forming a triangle.This characterization implies the value of \(\sigma(K_6 – E(K_3), n)\).

We propose and study game-theoretic versions of independence
in graphs. The games are played by two players – the aggressor and the
defender – taking alternate moves on a graph G with tokens located on
vertices from an independent set of \(G\). A move of the aggressor is to select
a vertex v of \(G\). A move of the defender is to move tokens located on
vertices in \(N_G(v)\) each along one incident edge. The goal of the defender is
to maintain the set of occupied vertices independent while the goal of the
aggressor is to make this impossible. We consider the maximum number of
tokens for which the aggressor can not win in a strategic and an adaptive
version of the game.

In this study, we investigate Diophantine equations using the generalized Fibonacci and Lucas sequences. We obtain all integer solutions for several Diophantine equations such as \(x^2 -kxy- y^2 = \mp 1,\) \(x^2 -kxy+ y^2 = 1,\) \(x^2 – kxy-y^2 = \mp (k^2+4),\)
\(x^2 – (k^2 + 4)xy + (k^2+4)y^2 =\mp k^2,\) \(x^2 – kxy +y^2 = -(k^2-4)\). and \(x^2-(k^2-4)xy-(k^2-4)y^2=k^2\)
Some of these results are previously known, but we provide new and distinct proofs using generalized Fibonacci and Lucas sequences.

Let \(G = \{g_1, \ldots, g_n\}\) be a finite abelian group. Consider the complete graph \(K_n\) with vertex set \(\{g_1, \ldots, g_n\}\). A \(G\)-coloring of \(K_n\) is a proper edge coloring where the color of edge \(\{g_i, g_j\}\) is \(g_i + g_j\), \(1 \leq i 2\), there exists a proper edge coloring of \(K_p\) which is decomposable into multicolored Hamilton cycles.

It is shown that \(r(K_{1,m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{m+1}\) for any two fixed integers \(k \geq m \geq 2\) and \(n \to \infty\).
This result is obtained using the analytic method and the function \(f_{m}(x) = \int_0^1 \frac{(1-t)^{\frac{1}{m}}dt}{m+(x-m)^t} , \quad x \geq 0,m \geq 1,\)
building upon the upper bounds for \(r(K_{m,k}, K_n)\) established by Y. Li and W. Zang.Furthermore, \((c – o(1)) (\frac{n}{log n})^{\frac{7}{3}}\leq r(W_{4}, K_n) \leq (1 + o(1)) (\frac{n}{log n})^{3}\) (as \(n \to \infty\)). Moreover, we derive
\(r(K_{1} + K_{m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{l+m}\) for any two fixed integers \(k \geq m \geq 2\) (as \(n \to \infty\)).

A simple graph \(G = (V, E)\) admits an \(H\)-covering if every edge in \(E\) belongs to a subgraph of \(G\) isomorphic to \(H\). We say that \(G\) is \(H\)-magic if there exists a total labeling \(f: V \cup E \rightarrow \{1, 2, \ldots, |V| + |E| + 1\}\) such that for each subgraph \(H’ = (V’, E”)\) of \(G\) isomorphic to \(H\),
\(\sum_{v \in V’} f(v) + \sum_{e \in E”} f(e)\)
is constant.

When \(f(V) = \{1, 2, \ldots, |V|\}\), then \(G\) is said to be \(H\)-supermagic.

In this paper, we show that all prism graphs \(C_n \times P_m\), except for \(n = 4\), the ladder graph \(P_3 \times P_n\), and the grid \(P_3 \times P_n\), are \(C_4\)-supermagic.

The average crosscap number of a graph \(G\) is the expected value of the crosscap number random variable, over all labeled \(2\)-cell non-orientable embeddings of \(G\). In this study, some experimental results for average crosscap number are obtained. We calculate all average crosscap numbers of graphs with Betti number less than \(5\). As a special case, the smallest ten values of average crosscap number are determined. The distribution of average crosscap numbers of all graphs in \({R}\) is sparse. Some structure theorems for average crosscap number with a given or bounded value are provided. The exact values of average crosscap numbers of cacti and necklaces are determined. The crosscap number distributions of cacti and necklaces of type \((r,0)\) are proved to be strongly unimodal, and the mode of the embedding distribution sequence is upper-rounding or lower-rounding of its average crosscap number. Some open problems are also proposed.

A Roman dominating function of a graph \(G\) is a labeling \(f: V(G) \rightarrow \{0,1,2\}\) such that every vertex with label \(0\) has a neighbor with label \(2\). The Roman domination number \(\gamma_R(G)\) of \(G\) is the minimum of \(\sum_{v \in V(G)} f(v)\) over such functions. The Roman domination subdivision number \(sd_{\gamma R}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the Roman domination number.

In this paper, we prove that if \(G\) is a graph of order \(n \geq 4\) such that \(\overline{G}\) and \(G\) have connected components of order at least \(3\), then
\(sd_{\gamma R}(G) + sd_{\gamma R}(\overline{G}) \leq \left\lfloor \frac{n}{2} \right\rfloor + 3.\)

In \textit{Ars Comb.} \({84} (2007), 85-96\), Pedersen and Vestergaard posed the problem of determining a lower bound for the number of independent sets in a tree of fixed order and diameter \(d\). Asymptotically, we give here a complete solution for trees of diameter \(d \leq 5\). The lower bound is \(5^{\frac{n}{3}}\) and we give the structure of the extremal trees. A generalization to connected graphs is stated.