相关论文: Most Generalized Petersen graphs of girth 8 have c…

We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any…

We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber…

In this paper the strong metric dimension of generalized Petersen graphs $GP(n,2)$ is considered. The exact value is determined for cases $n=4k$ and $n=4k+2$, while for $n=4k+1$ an upper bound of the strong metric dimension is presented.

Due to their broad application to different fields of theory and practice, generalized Petersen graphs $GPG(n,s)$ have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for…

For natural numbers $n$ and $k$ ($n > 2k$), a generalized Petersen graph $P(n,k)$, is defined by vertex set $\lbrace u_i,v_i\rbrace$ and edge set $\lbrace u_iu_{i+1},u_iv_i,v_iv_{i+k}\rbrace$; where $i = 1,2,\dots,n$ and subscripts are…

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup…

It is known that the problem of computing the edge dimension of a graph is NP-hard, and that the edge dimension of any generalized Petersen graph $P(n,k)$ is at least 3. We prove that the graph $P(n,3)$ has edge dimension 4 for $n\ge 11$,…

In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when $G$ is graph of girth $g$ and the minimum degree $\delta \geq 2$, then $c(G) = O(n\log(n)(\delta-1)^{-\lfloor \frac{g+1}{4} \rfloor})$ as a…

The metric dimension of a graph $G$ is defined as the minimum number of vertices in a subset $S\subset V(G)$ such that all other vertices are uniquely determined by their distances to the vertices in $S$, and is denoted by $\dim(G)$. In…

The crossing number of a graph is the least number of crossings of edges among all drawings of the graph in the plane. In this article, we prove that the crossing number of the generalized Petersen graph P(10, 3) is equal to 6.

The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. The theoretical properties of the edge metric representations and the edge metric dimension of generalized Petersen graphs…

Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…

In any network, the interconnection of nodes by means of geodesics and the number of geodesics existing between nodes are important. There exists a class of centrality measures based on the number of geodesics passing through a vertex.…

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and…

The star chromatic index of a graph $G$, denoted by $\chi^\prime_s(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the…

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph…

In this paper, we define the $4$-girth-thickness $\theta(4,G)$ of a graph $G$ as the minimum number of planar subgraphs of girth at least $4$ whose union is $G$. We obtain the $4$-girth-thickness of the arbitrary complete graph $K_n$…

The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by…

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

The $4$-girth-thickness $\theta(4,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least four whose union is $G$. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is…