It was conjectured in a recently published paper that for any integer \(k \geq 8\) and any even integer \(n\) with \(2k+3 < n < 2k+\lfloor\frac{k}{2}\rfloor+3\), the \(k\)th power \(C_n^k\) of the \(n\)-cycle is not a divisor graph. In this paper, we prove this conjecture, hence obtaining a complete characterization of those powers of cycles which are divisor graphs.

Inspired by a recent paper by Giulietti, Korchmàros and Torres \([3]\), we provide equations for some quotient curves of the Deligne-Lusztig curve associated to the Suzuki group \(S_z(q)\).

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n-2l}}{b+c\prod\limits_{i=0}^{k+1}x_{n-2i}}, \quad n=0,1,\ldots,\]

where \(a\), \(b\), and \(c\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\) and \(l\) are nonnegative integers, with \(l \leq k+1\).

The chromatic polynomial of a graph \(\Gamma\), \(C(\Gamma; \lambda)\), is the polynomial in \(\lambda\) which counts the number of distinct proper vertex \(\lambda\)-colorings of \(\Gamma\), given \(\lambda\) colors. By applying the addition-contraction method, chromatic polynomials of some sequences of \(2\)-connected graphs satisfy a number of recursive relations. We will show that by knowing the chromatic polynomial of a few small graphs, the chromatic polynomial of each of these sequences can be computed by utilizing either matrices or generating functions.

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\). A simple graph \(G\) is of \(f\)-class 1 if the \(f\)-chromatic index of \(G\) equals \(\Delta_f(G)\), where \(\Delta_f(G) = \max_{v\in V(G)}\{\left\lceil\frac{d(v)}{f(v)}\right\rceil\}\). In this article, we find a new sufficient condition for a simple graph to be of \(f\)-class 1, which is strictly better than a condition presented by Zhang and Liu in 2008 and is sharp. Combining the previous conclusions with this new condition, we improve a result of Zhang and Liu in 2007.

We provide the specifics of how affine planes of orders three, four, and five can be used to partition the full design comprising all triples on \(9, 16\), and \(25\) elements, respectively. Key results of the approach for order five are generalized to reveal when there is potential for using suitable affine planes of order \(n\) to partition the complete sets of \(n^2\) triples into sets of mutually disjoint triples covering either all \(n^2\), or else precisely \(n^2 – 1\), elements.

In this paper, the notion of left-right and right-left \(f\)-derivation of a BCC-algebra is introduced, and some related properties are investigated. Also, we consider regular \(f\)-derivation and \(d\)-invariant on \(f\)-ideals in BCC-algebras.

Let \(G\) be a planar graph with maximum degree \(\Delta\). It’s proved that if \(\Delta \geq 5\) and \(G\) does not contain \(5\)-cycles and \(6\)-cycles, then \(la(G) = \lceil\frac{\Delta(G)}{2}\rceil\).

We call the digraph \(D\) an \(m\)-coloured digraph if the arcs of \(D\) are coloured with \(m\) colours. A subdigraph \(H\) of \(D\) is called monochromatic if all of its arcs are coloured alike.

A set \(N \subseteq V(D)\) is said to be a kernel by monochromatic paths if it satisfies the following two conditions:

(i) For every pair of different vertices \(u,v \in N\) there is no monochromatic directed path between them.

(ii) For every vertex \(x \in V(D) – N\), there is a vertex \(y \in N\) such that there is an \(xy\)-monochromatic directed path.

In this paper, it is proved that if \(D\) is an \(m\)-coloured \(k\)-partite tournament such that every directed cycle of length \(3\) and every directed cycle of length \(4\) is monochromatic, then \(D\) has a kernel by monochromatic paths.

Some previous results are generalized.

Let \(\mathcal{S}\) be a finite family of sets in \(\mathbb{R}^d\), each a finite union of polyhedral sets at the origin and each having the origin as an extreme point. Fix \(d\) and \(k\), \(0 \leq k \leq d \leq 3\). If every \(d+1\) (not necessarily distinct) members of \(\mathcal{S}\) intersect in a star-shaped set whose kernel is at least \(k\)-dimensional, then \(\cap\{S_i:S_i\in\mathcal{S}\}\) also is a star-shaped set whose kernel is at least \(k\)-dimensional. For \(k\neq 0\), the number \(d+1\) is best possible.