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 \(C_n\).

A vertex cut that separates the connected graph into components such that every vertex in these components has at least \(g\) neighbors is an \(R^g\)-vertex-cut. \(R^g\)-vertex-connectivity, denoted by \(\kappa^g(G)\), is the cardinality of a minimum \(R^g\)-vertex-cut of \(G\). In this paper, we will determine \(\kappa^g\) and characterize the \(R^g\)-vertex-atom-part for the first and second type Harary graphs.

A graph \(G\) is supereulerian if \(G\) has a spanning eulerian subgraph. We use \(\mathcal{SL}\) to denote the families of supereulerian graphs. In 1995, Zhi-Hong Chen and Hong-Jian Lai presented the following open problem [2, problem 8.8]: Determine

\[L=\min\max\limits_{G\in SL-\{K_1\}}\{\frac{|E(H)|}{|E(G)|} : H \text{ is spanning eulerian subgroup of G}\}.\]

For a graph \(G\), \(O(G)\) denotes the set of all odd-degree vertices of \(G\). Let \(G\) be a simple graph and \(|O(G)| = 2k\). In this note, we show that if \(G\in{SL}\) and \(k \leq 2\), then \(L \geq \frac{2}{3}\).

It is known that the number of Dyck paths is given by a Catalan number. Dyck paths are represented as plane lattice paths which start at the origin \(O\) and end at the point \(P_n = (n,n)\) repeating \((1,0)\) or \((0,1)\) steps without going above the diagonal line \(OP_n\). Therefore, it is reasonable to ask of any positive integers \(a\) and \(b\) what number of lattice paths start at \(O\) and end at point \(A = (a, b)\) repeating the same steps without going above the diagonal line \(OA\). In this article, we show a formula to represent the number of such generalized Dyck paths.