The automorphic A-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given subgroup A of the full automorphism group of Γ. We compute the automorphic A-chromatic index of each generalized Petersen graph when A is the full automorphism group.

A perfectly one-factorable (P1F) regular graph G is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. We consider the case in which G is a cubic graph. The existence of a P1F cubic graph is guaranteed for each admissible value of the number of vertices. We give conditions for determining P1F graphs within a subfamily of...

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 3-regular 4-ordered graphs other than K4 and K3,3. In 2008, Meszaros solve the question by proving the Petersen graph and the Heawood graph are 3-regular 4-ordered gr...

Robertson ([5]) and independently, Bondy ([1]) proved that the generalized Petersen graph P (n, 2) is non-hamiltonian if n ≡ 5 (mod 6), while Thomason [7] proved that it has precisely 3 hamiltonian cycles if n ≡ 3 (mod 6). The hamiltonian cycles in the remaining generalized Petersen graphs were enumerated by Schwenk [6]. In this note we give a short unified proof of these results using Grinberg...

Determining the size of minimum vertex cover of a graph G, denoted by β(G), is an NP-complete problem. Also, for only few families of graphs, β(G) is known. We study the size of minimum vertex cover in generalized Petersen graphs. For each 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...

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph. ”The Petersen graph is an obstruction to many properties in graph theory, and often is, or conjectured to be, the only obstruction”. This phrase is taken from one of the series of papers by Robertson, Sanders, Seymour and Thomas that is devoted to the ...

1 Department of Mathematics, Lahore College for Women University, Lahore-Pakistan E-mail: [email protected] 2 Department of Mathematics, Lahore College for Women University, Lahore-Pakistan E-mail: [email protected] 3* Division of Science and Technology, University of Education, Township Lahore-Pakistan E-mail: [email protected] ABSTRACT: Let G be a connected graph with diameter...

Abstract A graph G is $$\alpha $$ α -excellent if every vertex of contained in some maximum independent set . In this paper, we characterize bipartite graphs, unicyclic simplicial chordal block and show that generalized Petersen -excellent.