In \(2006\), Mojdeh and Jafari Rad [On the total domination critical graphs, Electronic Notes in Discrete Mathematics, 24 (2006), 89-92] gave an open problem: Does there exist a \(3\)-\(\gamma_t\)-critical graph \(G\) of order \(\Delta(G) + 3\) with \(\Delta(G)\) odd and \(\delta(G) \geq 2\)? In this paper, we positively answer that for each odd integer \(n \geq 9\), there exists a \(3\)-\(\gamma_t\)-critical graph \(G_n\) of order \(n+3\) with \(\delta(G) \geq 2\). On the contrary, we also prove that for \(\Delta(G) = 3, 5, 7\), there is no \(3\)-\(\gamma_t\)-critical graph of order \(\Delta(G) + 3\) with \(\delta(G) \geq 2\).

Let \(\{w_n\}\) be a second-order recurrence sequence. According to the definition and characteristics of the recurrent sequence, we proved a recursion formula for certain reciprocal sums whose denominators are products of consecutive elements of \(\{w_n\}\).

Let \(G\) be a graph in which each vertex has been colored using one of \(k\) colors, say \(c_1, c_2, \ldots, c_k\). If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices colored \(c_i\), \(i = 1, 2, \ldots, k\), and \(|n_i – n_j| \leq 1\) for any \(i, j \in \{1, 2, \ldots, k\}\), then \(C\) is equitably \(k\)-colored. An \(m\)-cycle decomposition \(\mathcal{C}\) of a graph \(G\) is equitably \(k\)-colorable if the vertices of \(G\) can be colored so that every \(m\)-cycle in \(\mathcal{C}\) is equitably \(k\)-colored. For \(m = 4, 5\), and \(6\), we completely settle the existence problem for equitably \(2\)-colorable \(m\)-cycle decompositions of complete graphs with the edges of a \(1\)-factor added.

Suppose a network facility location problem is modelled by means of an undirected, simple graph \(G = (\mathcal{V, E})\) with \(\mathcal = \{v_1, \ldots, v_n\}\). Let \(\mathbf{r} = (r_1, \ldots, r_n)\) and \(\mathbf{s} = (s_1, \ldots, s_n)\) be vectors of nonnegative integers and consider the combinatorial optimization problem of locating the minimum number, \(\gamma(\mathbf{r}, \mathbf{s}, G)\) (say), of commodities on the vertices of \(G\) such that at least \(s_j\) commodities are located in the vicinity of (i.e. in the closed neighbourhood of) vertex \(v_j\), with no more than \(r_j\) commodities placed at vertex \(v_j\) itself, for all \(j = 1, \ldots, n\). In this paper we establish lower and upper bounds on the parameter \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a general graph \(G\). We also determine this parameter exactly for certain classes of graphs, such as paths, cycles, complete graphs, complete bipartite graphs and establish good upper bounds on \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a class of grid graphs in the special case where \(r_j = r\) and \(s_j = s\) for all \(j = 1, \ldots, n\).

Let \(A\) be an arbitrary circulant stochastic matrix, and let \(\underline{x}_0\) be a vector. An “asymptotic” canonical form is derived for \(A^k\) (as \(k \to \infty\)) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with transition matrix \(A\), and by analyzing how \(A\) acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of \(A^k\) is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence \(\underline{x}_k = A^k \underline{x}_0\), is accomplished by utilizing congruential matrix identities and \((0,1)\)-matrices to determine the minimum \(2\)-adic order of the coordinates of \(\underline{x}_k\), through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeatedly applied simultaneously to each term of a finite sequence when read cyclically.

A graph \(G\) is \(s\)-Hamiltonian if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) has a Hamiltonian cycle, and \(s\)-Hamiltonian connected if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) is Hamiltonian-connected. Let \(k > 0, s \geq 0\) be two integers. The following are proved in this paper:(1) Let \(k \geq s+2\) and \(s \leq n-3\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s)/2\) for every independent set \(I\) of order \(k-s\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s-1\), then \(G\) is \(s\)-Hamiltonian.(2) Let \(k \geq s+3\) and \(s \leq n-2\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s+1)/2\) for every independent set \(I\) of order \(k-s-1\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s\), then \(G\) is \(s\)-Hamiltonian connected.These results extend several former results by Dirac, Ore, Fan, and Chen.

We show that every generalized quadrangle of order \((4,6)\) with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle \(\text{AS}(5)\). It then easily follows that every generalized quadrangle of order \(5\) with an axis of symmetry is isomorphic to the classical generalized quadrangle \(\text{Q}(4, 5)\).

A \(C_5C_7\) net is a trivalent decoration made by alternating pentagons \(C_5\) and heptagons \(C_7\). It can cover either a cylinder or a torus. In this paper, we compute the Szeged index of \(HC_5C_7[ r, p ]\) nanotube.

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types \(C, D\), and \(L\). Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.

The number of essentially different square polyominoes of order \(n\) and minimum perimeter \(p(n)\) is enumerated.