In \([V_2]\), Vince outlined three potential graph algorithms for \(S^3\) recognition, namely shelling, reducing, and closing; on the other hand, he conjectured that the graph \(H_0\ ) of Fig.1 – which proves the first two to fail – could be a counterexample for the third one, too. This note shows that the conjecture is false; so, the validity of the closing algorithm is still an open problem.

We consider two variations of the classical Ramsey number. In particular, we seek the number of vertices necessary to force the existence of an induced regular subgraph on a prescribed number of vertices.

The \(i\)-center \(C_i(G)\) of a graph \(G\) is the set of vertices whose distances from any vertex of \(G\) are at most \(i\). A vertex set \(X\) \(k\)-dominates a vertex set \(Y\) if for every \(y \in Y\) there is a \(x \in X\) such that \(d(x,y) \leq k\). In this paper, we prove that if \(G\) is a \(P_t\)-free graph and \(i \geq \lfloor\frac{t}{2}\rfloor \), then \(C_i(G)\) \((q+1)\)-dominates \(C_{i+q}(G)\), as conjectured by Favaron and Fouquet [4].

As a generalization of a matching consisting of edges only, Alavi et al. in [1] define a total matching which may contain both edges and vertices. Using total matchings, [1] defines a parameter \(\beta’_2(G)\) and proves that \(\beta’_2(G) \leq p-1\) holds for a connected graph of order \(p \geq 2\).
Our main result is to improve this inequality to \(\beta’_2(G) \leq p-2\sqrt{p}+{2}\) and we give an example demonstrating this bound to be best possible.
Relations of several other parameters to \(\beta’_2\) are demonstrated.

We examine permutations having a unique fixed point and a unique reflected point; such permutations correspond to permutation matrices having exactly one \(1\) on each of the two main diagonals. The permutations are of two types according to whether or not the fixed point is the same as the reflected point. We also consider permutations having no fixed or reflected points; these have been enumerated using two different methods, and we employ one of these to count permutations with unique fixed and reflected points.

Let \(G\) be a graph and \(t(G)\) be the number of triangles in \(G\). Define \(\mathcal G_n\) to be the set of all graphs on \(n\) vertices that do not contain a wheel and \(t_n = \max\{t(G) : G \in \mathcal G_n\}\).
T. Gallai conjectured that \(t_n \leq \lfloor\frac{n^2}{8}\rfloor\). In this note we describe a graph on \(n\) vertices that contains no wheel and has at least \(\frac{n^2+n}{8}-3\) triangles.

A construction is given which uses \(\text{PG}_i(d, q)\) and \(q\) copies of \(\text{AG}_i(d, q)\) to construct designs having the parameters of \(\text{PG}_{i+1}(d+1, q)\), where \(q\) is a prime power and \(i \leq d-1\).

In a previous paper, [6], we associated with every hyperoval of a projective plane of even order a Hadamard \(2\)–design and investigated when this design has lines with three points. We study further this problem using the concept of regular triple and prove the existence of lines with three points in Hadamard designs associated with translation hyperovals. In the general case, the existence of a secant line of regular triples implies that the order of the projective plane is a power of two.