The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Π1, . . . ,Πk, where Πi, i ∈ [k], is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then χρ(S(G)) ≤ 5, where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it ...

This thesis puts forward the conjecture that for n > 3k with k > 2, the generalized Petersen graph, GP (n, k) is Hamilton-laceable if n is even and k is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case n = 3k + 1 and k ≥ 1. We do this mainly by means of an induction which takes us from GP (3k + 1, k) to GP (3(k+2)+1, k+2). T...

Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.

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 vertextransitive 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 e...

In this paper, we show that if G is a 3-edge-connected graph with S V ðGÞ and jSj 12, then eitherG has an Eulerian subgraphH such that S V ðHÞ, or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3-edge-connected planar graph, then for any jSj 23, G has an Eulerian subgraph H such that S V...