A \(k\)-chromatic graph \(G\) is \(uniquely\) \(k\)-\(colorable\) if \(G\) has only one \(k\)-coloring up to permutation of the colors. In this paper, we focus on uniquely \(k\)-colorable graphs on surfaces. Let \({F}^2\) be a closed surface, excluding the sphere, and let \(\chi({F}^2)\) denote the maximum chromatic number of graphs embeddable on \({F}^2\). We shall prove that the number of uniquely \(k\)-colorable graphs on \({F}^2\) is finite if \(k \geq 5\), and characterize uniquely \(\chi({F}^2)\)-colorable graphs on \({F}^2\). Moreover, we completely determine uniquely \(k\)-colorable graphs on the projective plane for \(k \geq 5\).

Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of a graph, denoted by \(f(G)\), is the minimal integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we calculate the pebbling number of the graph \(D_{n,C_m}\) and consider the relationship the pebbling number between the graph \(D_{n,C_m}\) and the subgraphs of \(D_{n,C_m}\).

Let \(G\) and \(H\) be two graphs. A proper vertex coloring of \(G\) is called a dynamic coloring if, for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic coloring with \(k\) colors is denoted by \(\chi_2(G)\). We denote the Cartesian product of \(G\) and \(H\) by \(G \square H\). In this paper, we prove that if \(G\) and \(H\) are two graphs and \(\delta(G) \geq 2\), then \(\chi_2(G \square H) \leq \max(\chi_2(G), \chi(H))\). We show that for every two natural numbers \(m\) and \(n\), \(m, n \geq 2\), \(\chi_2(P_m \square P_n) = 4\). Additionally, among other results, it is shown that if \(3\mid mn\), then \(\chi_2(C_m \square C_n) = 3\), and otherwise \(\chi_2(C_m \square C_n) = 4\).

In \([1]\), Hosam Abdo and Darko Dimitrov introduced the total irregularity of a graph. For a graph \(G\), it is defined as
\[\text{irr}_t(G) =\frac{1}{2} \sum_{{u,v} \in V(G)} |d_G(u) – d_G(v)|,\]
where \(d_G(u)\) denotes the vertex degree of a vertex \(u \in V(G)\). In this paper, we introduce two transformations to study the total irregularity of unicyclic graphs and determine the graph with the maximal total irregularity among all unicyclic graphs with \(n\) vertices.

We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, \(et \;al\). The solution to the original problem is the well known Catalan numbers, while the variations discussed in this paper yield the Motzkin numbers and other related sequences. For this variation, we present a generating function for the sum of the labels on the balls.

A graph \(G\) of order \(n\) is called a tricyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n + 2\). Let \(\mathcal{T}_n\) denote the set of all tricyclic graphs on \(n\) vertices. In this paper, we determine the first to nineteenth largest Laplacian spectral radii among all graphs in the class \(\mathcal{T}_n\) (for \(n \geq 11\)), together with the corresponding graphs.

The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we determine the lower bounds for the Hosoya index of unicyclic graph with a given diameter. The corresponding extrenal graphs are characterized.

A subset \(S\) of vertices of a graph \(G\) is called a global connected dominating set if \(S\) is both a global dominating set and a connected dominating set. The global connected domination number, denoted by \(\gamma_{gc}(G)\), is the minimum cardinality of a global connected dominating set of \(G\). In this paper, sharp bounds for \(\gamma_{gc}\) are supplied, and all graphs attaining those bounds are characterized. We also characterize all graphs of order \(n\) with \(\gamma_{gc} = k\), where \(3 \leq k \leq n-1\). Exact values of this number for trees and cycles are presented as well.

Let \(\mathbb{F}_q^n\) denote the \(n\)-dimensional row vector space over the finite field \(\mathbb{F}_q\), where \(n \geq 2\). An \(l\)-partial linear map of \(\mathbb{F}_q^n\) is a pair \((V, f)\), where \(V\) is an \(l\)-dimensional subspace of \(\mathbb{F}_q^n\) and \(f: V \to \mathbb{F}_q^n\) is a linear map. Let \(\mathcal{L}\) be the set of all partial linear maps of \(\mathbb{F}_q^n\) containing \(1\). Ordered \(\mathcal{L}\) by ordinary and reverse inclusion, two families of finite posets are obtained. This paper proves that these posets are lattices, discusses their geometricity, and computes their characteristic polynomials.

A total coloring of a graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring \(h\) of a simple graph \(G = (V, E)\) is a proper total coloring of \(G\) such that \(H(u) \neq H(v)\) for any two adjacent vertices \(u\) and \(v\), where \(H(u) = \{h(wu) \mid wu \in E(G)\} \cup \{h(u)\}\) and \(H(v) = \{h(xv) \mid xv \in E(G)\} \cup \{h(v)\}\). The minimum number of colors required for a proper total coloring (resp. an adjacent vertex-distinguishing total coloring) of \(G\) is called the total chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of \(G\) and denoted by \(\chi_t(G)\) (resp. \(\chi_{at}(G)\)). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\), \(\chi(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2\). \(G\) is called Type 1 (resp. Type 2) if \(\chi_t(G) = \Delta(G) + 1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the augmented cube \(AQ_n\) is of Type 1 for \(n \geq 4\). We also consider the adjacent vertex-distinguishing total chromatic number of \(AQ_n\) and prove that \(\chi_{at}(AQ_n) = \Delta(AQ_n) + 2\) for \(n \geq 3 \).