A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 adjacent to at least one assigned 2 or 3, and with f(u)=0 3 two vertices 2. The weight of f equals sum w(f)=?v?Vf(v). minimum G called domination number ?dR(G) G. We obtain tight bounds in some cases closed expressions generalized Petersen graphs P(ck,k). In shor...

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...

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs P(ck,k). Exact values are established for all infinite families where general lower bound 45ck is attained. In other cases and upper bounds with small gaps given.

Isometric subgraphs of hypercubes are known as partial cubes. The subdivision graph of a graph G is obtained from G by subdividing every edge of G. It is proved that for a connected graph G its subdivision graph is a partial cube if and only if every block of G is either a cycle or a complete graph. Regular partial cubes are also considered. In particular it is shown that among the generalized ...

We consider the class of I-graphs I(n, j, k), which is a generalization over the class of the generalized Petersen graphs. We study different properties of I-graphs such as connectedness, girth and whether they are bipartite or vertex-transitive. We give an efficient test for isomorphism of I-graphs and characterize the automorphism groups of I-graphs. Regular bipartite graphs with girth at lea...

AgraphG is k-ordered if for every sequence of k distinct vertices ofG, there exists a cycle inG containing these k vertices in the specified order. It is k-ordered-Hamiltonian if, in addition, the required cycle is a Hamiltonian cycle in G. The question of the existence of an infinite class of 3-regular 4-ordered-Hamiltonian graphs was posed in Ng and Schultz in 1997 [2]. At the time, the only ...