Let \(G\) be an undirected graph. An orientation \(\vec G\) of \(G\) is a directed graph in which every edge of \(G\) is assigned a direction. Directed graph burning is an iterative graph exploration process in which each round consists of two phases: spreading, then vertex selection. Each vertex is initially considered unburned. In the spreading phase, each burned vertex burns all of its out-neighbors. In the vertex selection phase, a vertex is chosen and burned. The burning number of a directed graph \(b(\vec G)\) is the minimum number of rounds necessary to burn all vertices of \(G\). The orientable burning number of an undirected graph \(G\), denoted \(B(G)\), is the maximum burning number across all orientations of \(G\). In this paper, we prove two main results. First, for any integer \(k\) between the minimum and maximum burning numbers of orientations of \(G\), there exists an orientation \(\vec G\) such that \(b(\vec G) = k\). We also determine the orientable burning number of wheels using a bound on the join of directed graphs.

A gap-\([q]\)-vertex labelling of a graph \(G\) is an assignment \(f:V(G)\to\{1,\ldots,q\}\) for which the gap between the largest and smallest labels in the neighbourhood of each vertex induces a proper vertex colouring. The minimum such \(q\) is the vertex-gap number \(\chi_v^g(G)\). We investigate this parameter for generalized Petersen graphs \(P(n,k)\). First, we prove that every bipartite generalized Petersen graph, equivalently every \(P(n,k)\) with \(n\) even and \(k\) odd, has vertex-gap number \(2\); this gives a complete characterization of the generalized Petersen graphs admitting a gap-\([2]\)-vertex labelling. We then construct explicit gap-\([3]\)-vertex labellings for several non-bipartite classes, including families determined by congruence conditions modulo \(4\), \(5\), and \(10\), the family \(n=2tk\) with even \(k\), the cases \(k=1\) and \(k=2\) for specified odd orders, and the family \(n=3k\) with \(k\geq3\). In each case the lower bound follows from the chromatic number, while the upper bound is established by an explicit labelling and a verification of the induced colours on the three edge types of \(P(n,k)\). The results support a natural conjecture that, apart from the exceptional graphs \(P(3,1)\) and \(P(6,2)\), every non-bipartite generalized Petersen graph has vertex-gap number \(3\).

The deck of a topological space \( X \), denoted \( \mathcal{D}(X) = \{[X_x] : x \in X\} \), comprises the homeomorphism classes of subspaces obtained by removing a single point \( x \) from \( X \). A space \( X \) is  topologically reconstructible if \( \mathcal{D}(X) = \mathcal{D}(Y) \) implies that \( X \) is homeomorphic to \( Y \). The \( n \)-deck of \( X \), defined as \( \mathcal{D}_n(X) = \{[X – \{x_1, x_2, \dots, x_n\}] : x_1, x_2, \dots, x_n \in X\} \), generalizes this concept to the removal of \( n \) points. A space \( X \) is topologically \( n \)-reconstructible if \( \mathcal{D}_n(X) = \mathcal{D}_n(Y) \) implies \( X \cong Y \). The multi \( n \)-deck of \( X \) includes all \( n \)-cards, representing every possible homeomorphism type. A space is weakly \( n \)-reconstructible if it can be uniquely determined from its multi \( n \)-deck. This paper establishes that certain hyper-connected topological spaces are weakly 2-reconstructible.

Relying on the Classification of Finite Simple Groups it was shown by Feng and Xu (Discrete Math., 2005) that every quartic Cayley graph of a regular \(p\)-group, \(p \neq 2,5\), is normal. In this paper a CFSG-free proof of a special case of Feng-Xu theorem is given. Along the way it is also proved that for an arbitrary \(p\)-group \(G\) with a minimum set \(\{a,b\}\) of two generators, one of which is of order \(p\), in the corresponding Cayley graph \(\mathrm{Cay}(G,\{a,a^{-1},b,b^{-1}\})\) the induced action of vertex stabilizer on the neighbors’ set is contained in the dihedral group \(D_8\).

The renowned Gossiping Problem (1971) asks the following. There are \(n\) people who each know an item of gossip. In a telephone call, two people share all the gossip they know. How many calls are needed for all of them to be informed of all the gossip? If \(n\ge 4\), the answer is \(2n-4\). We initiate and solve the related Greedy Gossiping Problem: given a fixed number \(m<2n-4\) of calls, at most how much gossip can be known altogether? Our main result is that if every call increases the total knowledge of gossip as much as possible, then this call strategy is optimal.