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.

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The second power of paths \(P_n^2\),is the graph obtained from the path \(P_n\) by adding edges that join all vertices \(u\) and \(v\) with \(d(u,v) = 2\). In this paper, we show that certain combinations of second power of paths, paths, cycles, and stars are cordial. Specifically, we investigate the cordiality of the join and the union of pairs of second power of paths and graphs consisting of one second power of path and one path and one cycle.

We initiate a study of the toughness of infinite graphs by considering a natural generalization of that for finite graphs. After providing general calculation tools, computations are completed for several examples. Avenues for future study are presented, including existence problems for tough-sets and calculations of maximum possible toughness. Several open problems are posed.

Let an \(H\)-point be a vertex of a tiling of \(\mathbb{R}^2\) by regular hexagons of side length 1, and \(D(n)\) a circle of radius \(n\) (\(n \in \mathbb{Z}^+\)) centered at an \(H\)-point. In this paper, we present an algorithm to calculate the number, \(\mathcal{N}_H(D(n))\), of H-points that lie inside or on the boundary of \(D(n)\). Furthermore, we show that the ratio \(\mathcal{N}_H(D(n))/n^2\) tends to \(\frac{2\pi}{S}\) as \(n\) tends to \(\infty\), where \(S = \frac{3\sqrt{3}}{2}\) is the area of the regular hexagonal tiles.

Let \(G\) be a finite, simple graph. We denote by \(\gamma(G)\) the domination number of \(G\). The bondage number of \(G\), denoted by \(b(G)\), is the minimum number of edges of \(G\) whose removal increases the domination number of \(G\). \(C_n\) denotes the cycle of \(n\) vertices. For \(n \geq 5\) and \(n \neq 5k + 3\), the domination number of \(C_5 \times C_n\) was determined in [6]. In this paper, we calculate the domination number of \(C_5 \times C_n\) for \(n = 5k + 3\) (\(k \geq 1\)), and also study the bondage number of this graph, where \(C_5 \times C_n\) is the cartesian product of \(C_5\) and \(C_n\).