Let \(G\) be a simple graph of order \(p \geq 2\). A proper \(k\)-total coloring of a simple graph \(G\) is called a \(k\)-vertex distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The notation \(\chi_{vt}(G)\) indicates the smallest number of colors required for which \(G\) admits a \(k\)-VDTC with \(k \geq \chi_{vt}(G)\). For every integer \(m \geq 3\), we will present a graph \(G\) of maximum degree \(m\) such that \(\chi_{vt}(G) < \chi_{vt}(H)\) for some proper subgraph \(H \subseteq G\).

Let \(G = (V,E)\) be a graph. Let \(\gamma(G)\) and \(\gamma_t(G)\) be the domination and total domination number of a graph \(G\), respectively. The \(\gamma\)-criticality and \(\gamma_t\)-criticality of Harary graphs are studied. The Question \(2\) of the paper [W. Goddard et al., The Diameter of total domination vertex critical graphs, Discrete Math. \(286 (2004), 255-261]\) is fully answered with the family of Harary graphs. It is answered to the second part of Question \(1\) of that paper with some Harary graphs.

Let \(G\) be a connected graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2} \sum_{u,v \in V(G)} d^2(u,v),\) with the summation going over all pairs of vertices in \(G\) and \(d(u,v)\) denotes the distance between \(u\) and \(v\) in \(G\). In this paper, we determine the upper or lower bounds on hyper-Wiener index of trees with given number of pendent vertices, matching number, independence number, domination number, diameter, radius, and maximum degree.

A large set of resolvable Mendelsohn triple systems of order \(v\), denoted by \(\text{LRMTS}(v)\), is a collection of \(v-2\) \(\text{RMTS}(v)\)s based on \(v\)-set \(X\), such that every Mendelsohn triple of \(X\) occurs as a block in exactly one of the \(v-2\) \(\text{RMTS}(v)\)s. In this paper, we use \(\text{TRIQ}\) and \(\text{LR-design}\) to present a new product construction for \(\text{LRMTS}(v)\)s. This provides some new infinite families of \(\text{LRMTS}(v)\)s.

In this paper, we investigate the existence of nontrivial solutions for the equation \(y(G \Box H) – \gamma(G) \gamma(H)\) fixing one factor. For the complete bipartite graphs \(K_{m,n}\), we characterize all nontrivial solutions when \(m = 2, n \geq 3\) and prove the nonexistence of solutions when \(m \geq 2, n \leq 3\). In addition, it is proved that the above equation has no nontrivial solution if \(A\) is one of the graphs obtained from \(G\), the cycle of length \(n\), either by adding a vertex and one pendant edge joining this vertex to any vertex to any \(v\in V(C_n)\), or by adding one chord joining two alternating vertices of \(C_n\).

For a graph \(G = (V(G), E(G))\), let \(i(G)\) be the number of isolated vertices in \(G\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\) if \(G\) is not complete; \(I(G) = |V(G)|-1\) otherwise. In this paper, several sufficient conditions in terms of isolated toughness are obtained for the existence of \([a, b]\)-factors avoiding given subgraphs, e.g., a set of vertices, a set of edges and a matching, respectively.

In a graph \(G\), the distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining them. The eccentricity \(e(u)\) of a vertex \(u\) is the distance to a vertex farthest from \(u\). The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph \(R(G)\) based on \(G\) has the vertex set as in \(G\). Two vertices \(u\) and \(v\) are adjacent in \(R(G)\) if the distance between them in \(G\) is equal to the radius of \(G\). If \(G\) is disconnected, then two vertices are adjacent in \(R(G)\) if they belong to different components. The main objective of this paper is to find a necessary and sufficient condition for a graph to be a radial graph.

Let \(\{T, T’\}\) be a Latin bitrade. Then \(T\) (and \(T’\)) is said to be \((r,c,e)\)-homogeneous if each row contains precisely \(r\) entries, each column contains precisely \(c\) entries, and each entry occurs precisely \(e\) times. An \((r,c,e)\)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely \((r,c,e) = (3,3,3), (4,4,2)\), or \((6,3,2)\). The first case has been completely classified by a number of authors. We present classifications for the other two cases.

In this paper, we prove an interesting property of rook polynomials for \(2\)-D square boards and extend that for rook polynomials for \(3\)-D cubic, and \(r\)-D “hypercubic” boards. In particular, we prove that for \(r\)-D rook polynomials the modulus of the sum of their roots equals their degree. We end with some further questions, mainly for the \(2\)-D and \(3\)-D case, that could serve as future projects.

Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\), then the subgraph \(H\) is called a \({spanning \;subgraph}\) of \(G\). A spanning subgraph \(H\) of \(G\) is called an \({F-factor}\) if each component of \(H\) is isomorphic to \(F\). Further, if there exists a subgraph of \(G\) whose vertex set is \(\lambda V(G)\) and can be partitioned into \(F\)-factors, then it is called a \({\lambda-fold \;F-factor}\) of \(G\), denoted by \(S_\lambda(1,F,G)\). A \({large \; set}\) of \(\lambda\)-fold \(F\)-factors in \(G\) is a partition \(\{\mathcal{B}_i\}_{i}\) of all subgraphs of \(G\) isomorphic to \(F\), such that each \((X, \mathcal{B}_i)\) forms a \(\lambda\)-fold \(F\)-factor of \(G\). In this paper, we investigate the large set of \(\lambda\)-fold \(P_3\)-factors in \(K_{v,v}\) and obtain its existence spectrum.