Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.

Using voltage graphs, we construct the the smallest known trivalent graph of girth 16. The graph has 960 vertices and is a lift of the Petersen graph. By way of preparation, other lifts of the Petersen graph are discussed. AMS Subject Classifications: 05C25, 05C35

A Roman domination function on a graph G = (V,E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman domination function f is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by...

An α-labeling of a bipartite graph Γ of size e is an injective function f : V (Γ) → {0, 1, 2, . . . , e} such that {|f(x) − f(y)| : [x, y] ∈ E(Γ)} = {1, 2, . . . , e} and with the property that its maximum value on one of the two bipartite sets does not reach its minimum on the other one. We prove that the generalized Petersen graph P8n,3 admits an α-labeling for any integer n ≥ 1 confirming th...

A graph G with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1, 2, . . . , |V | such that for every edge xy in E, the labels assigned to x and y are relatively prime or coprime. A graph is called prime if it has a prime labeling. In this paper, we show that generalized Petersen graphs P (n, 3) are not prime for odd n, prime for even n ≤ 100 ...

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G K2. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is sho...

A graph G is said to be distance-balanced if for any edge uv of G, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Let GP (n, k) be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer k > 2, there exists a positive integer n0 such that the GP (n, k...

The forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The forcing spectrum of G is the collection of forcing numbers of all perfect matchings of G. In this paper, we classify the perfect matchings of a generalized Petersen graph P (n, 2) in two types, and show that the forcing spectrum is the un...

In my doctoral thesis [1], I claimed a proof that the branchwidth of the Petersen graph had branchwidth 5 and proceeded to prove that it was a member of the obstruction set for graphs with branchwidth at most 4. I have stated this claim in Hicks [2] and Hicks et al. [4]. There was, however, an error in the construction of the tangle used to prove the claim. The branchwidth of the Petersen graph...