Let f be a function that assigns to each vertex a subset of colors chosen from a set C = {1, 2, . . . , k} of k colors. If  u∈N(v) f (u) = C for each vertex v ∈ V with f (v) = ∅, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u ∈ V | uv ∈ E}. The weight of f , denoted by w(f ), is defined as w(f ) =  v∈V |f (v)|. Given a graph G, the minimum weight among all weight...

A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise nonisomophic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments.

By strengthening an edge-decomposition technique for gracefully labelling a generalised Petersen graph, we provide graceful labellings for a new infinite family of such graphs. The method seems flexible enough to provide graceful labellings for many other classes of graphs in the future.

A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise nonisomophic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments.

It is well known that Julius Petersen’s famous paper on graph factorisation [9] has its origin in a problem which arose in connection with Hilbert’s proof of the Finite Basis Theorem for the invariants of binary forms [3]. Once one accepts the graph theoretic framework, the reformulation of the algebraic problem is straightforward and needs only a minimum of explanation-at least in the context ...

A graphG is k-ordered if for any sequence of k distinct vertices ofG, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 [10]. At the time, the only known examples were K4 and K3,...