The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence \(5\) and for general vertices with valence \(6\), with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.

This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the \(k\)th order between components, \(p\)-competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and \(m\)-step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.

A graph \( X \) is \( k \)-spanning cyclable if for any subset \( S \) of \( k \) distinct vertices there is a 2-factor of \( X \) consisting of \( k \) cycles such that each vertex in \( S \) belongs to a distinct cycle. In this paper, we examine the \( k \)-spanning cyclability of 4-valent Cayley graphs on Abelian groups.

A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of \( V \) and \( W \). These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\), which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.

For a graph \( G \) and for non-negative integers \( p, q \) and \( r \), the triplet \( (p, q, r) \) is said to be an admissible triplet, if \( 3p + 4q + 6r = |E(G)| \). If \( G \) admits a decomposition into \( p \) cycles of length \( 3 \), \( q \) cycles of length \( 4 \), and \( r \) cycles of length \( 6 \) for every admissible triplet \( (p, q, r) \), then we say that \( G \) has a \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition. In this paper, the necessary conditions for the existence of \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition of \( K_{\ell, m, n}(\ell \leq m \leq n) \) are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths \( 3 \) and \( 6 \), Austral. J. Combin., 73(1), 220-241 (2019)}.

For a graph \( G \) and a subgraph \( H \) of a graph \( G \), an \( H \)-decomposition of the graph \( G \) is a partition of the edge set of \( G \) into subsets \( E_i \), \( 1 \leq i \leq k \), such that each \( E_i \) induces a graph isomorphic to \( H \). In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the \( \lambda \)-fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

We classify the geometric hyperplanes of the Segre geometries, that is, direct products of two projective spaces. In order to do so, we use the concept of a generalised duality. We apply the classification to Segre varieties and determine precisely which geometric hyperplanes are induced by hyperplanes of the ambient projective space. As a consequence we find that all geometric hyperplanes are induced by hyperplanes of the ambient projective space if, and only if, the underlying field has order \(2\) or \(3\).

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

The hub cover pebbling number, \(h^{*}(G)\), of a graph $G$, is the least non-negative integer such that from all distributions of \(h^{*}(G)\) pebbles over the vertices of \(G\), it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for \(G\) using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.