The exact domination number of the generalized Petersen graphs

Let G=(V,E) be a graph. A subset S⊆V is a dominating set of G, if every vertex u∈V−S is dominated by some vertex v∈S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2008) 603-610] proved that and conjectured that the upper bound is the exact domination number. In this paper we prove this conjecture.     

On the domination number of the generalized Petersen graphs

Graph domination numbers and algorithms for finding them have been investigated for numerous classes of graphs, usually for graphs that have some kind of tree-like structure. By contrast, we study an infinite family of regular graphs, the generalized Petersen graphs G(n). We give two procedures that between them produce both upper and lower bounds for the (ordinary) domination number of G(n), and we conjecture that our upper bound ⌈3n/5⌉ is the exact domination number. To our knowledge this is one of the first classes of regular graphs for which such a procedure has been used to estimate the domination number.     

Domination in generalized Petersen graphs

Generalized Petersen graphs are certain graphs consisting of one quadratic factor. For these graphs some numerical invariants concerning the domination are studied, namely the domatic number , the total domatic number and the -ply domatic number for and . Some exact values and some inequalities are stated.     

Generalizing the generalized Petersen graphs

The generalized Petersen graphs (GPGs) which have been invented by Watkins, may serve for perhaps the simplest nontrivial examples of “galactic” graphs, i.e. those with a nice property of having a semiregular automorphism. Some of them are also vertex-transitive or even more highly symmetric, and some are Cayley graphs. In this paper, we study a further extension of the notion of GPGs with the emphasis on the symmetry properties of the newly defined graphs.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

The isomorphism classes of the generalized Petersen graphs

In earlier work involving cycles in Generalized Petersen Graphs, we noticed some unexpected instances of P(m,k)≅P(m,l). In this article, all such instances are characterized. A formula is presented for the number of isomorphism classes of P(m,k).     

On the Hamilton connectivity of generalized Petersen graphs

We investigate the Hamilton connectivity and Hamilton laceability of generalized Petersen graphs whose internal edges have jump 1, 2 or 3.     

On generalized Petersen graphs labeled with a condition at distance two

An L(2,1)-labeling of graph G is an integer labeling of the vertices in V(G) such that adjacent vertices receive labels which differ by at least two, and vertices which are distance two apart receive labels which differ by at least one. The λ-number of G is the minimum span taken over all L(2,1)-labelings of G. In this paper, we consider the λ-numbers of generalized Petersen graphs. By introducing the notion of a matched sum of graphs, we show that the λ-number of every generalized Petersen graph is bounded from above by 9. We then show that this bound can be improved to 8 for all generalized Petersen graphs with vertex order >12, and, with the exception of the Petersen graph itself, improved to 7 otherwise.     

Colouring graphs with bounded generalized colouring number

Given a graph G and a positive integer p, χ_p(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χ_p(G) in terms of the k-colouring number of G for k=2^p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χ_k+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:

(a)

For every positive integer p, χ_p(G) is bounded by a constant for all G∈K.

(b)

For every positive integer k, is bounded by a constant for all G∈K.

It was proved by Nešet?il and Ossona de Mendez that (a) is equivalent to the following:

(c)

For every positive integer q, ∇_q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.

Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing ∇_q−(1/2)(G) and by showing that there is a function F_k such that . This gives an alternate proof of the equivalence of (a) and (c).     

The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

Supereulerian graphs and the Petersen graph

A graphG is supereulerian if G has a spanning eulerian subgraph.Boesch et al.[J.Graph Theory,1,79–84(1977)]proposed the problem of characterizing supereulerian graphs.In this paper,we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph.This extends a former result of Catlin and Lai[J.Combin.Theory,Ser.B,66,123–139(1996)].     

Homomorphisms from sparse graphs to the Petersen graph

Let G be a graph and let c: be an assignment of 2-elements subsets of the set {1,…,5} to the vertices of G such that for any two adjacent vertices u and v,c(u) and c(v) are disjoint. Call such a coloring c a (5, 2)-coloring of G. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph.The maximum average degree of G is defined as . In this paper, we prove that every triangle-free graph with is homomorphic to the Petersen graph. In other words, such a graph is (5, 2)-colorable. Moreover, we show that the bound on the maximum average degree in our result is best possible.