For natural numbers n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and subscripts are reduced modulo n. Here first, we characterize minimum vertex covers in generalized Petersen graphs. Second, we present a lower bound and some upper bounds for β(P (n, k)), the size of minimum vertex cover...

A/Mmcr-It is proved that the Shannon zero-error capacity of the pentagon is e. The method is then generalized to obtain upper bounds on the capacity of au arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special g...

In 1987, Lovv asz conjectured that every brick G diierent from K 4 , C 6 , and the Petersen graph has an edge e such that G e is a matching covered graph with exactly one brick. Lovv asz and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. We present here an independent proof of their theorem. We shall in fact prove that if G is any brick diierent from K 4...

Assume that n and k are positive integers with n ≥ 2k + 1. A non-hamiltonian graph G is hypo hamiltonian if G − v is hamiltonian for any v ∈ V (G). It is proved that the generalized Petersen graph P (n, k) is hypo hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). Similarly, a hamiltonian graph G is hyper hamiltonian if G−v is hamiltonian for any v ∈ V (G). In this paper, we will give some nec...

The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices regular polygon to corresponding star polygon. This has many interesting properties. As result, it been widely researched. In this work, edge metric dimensions GP(2l + 1, l) and GP(2l, explored, is shown that dimension equal its dimension. Furthermore, proved upper bound same as value for l).

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K4 and the complete bipartite graph K3,3. In 2008, Meszaros solved the question by proving that the Petersen ...

Chartrand, G., H. Hevia and R.J. Wilson, The ubiquitous Petersen graph, Discrete Mathematics 100 (1992) 303-311.

A connected graph G = (V, E) is said to be (a, d)antimagic, for some positive integers a and d, if its edges admit a labeling by all the integers in the set {1, 2, . . . , |E(G)|} such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term a and the common difference d. Mirka Miller and Mar...