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For a permutation π of the vertex set of a graph G, the graph πG is obtained from two disjoint copies G 1 and G 2 of G by joining each v in G 1 to π(v) in G 2. Hence if π = 1, then πG = K 2 × G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2γ(G). We study graphs for which γ(K 2 × G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V (G)… (More)

A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is traceable. The Traceability Conjecture (TC) states that for k 2 every k-traceable oriented graph of order at least 2k − 1 is traceable. It has been shown that for 2 k 6, every k-traceable oriented graph is traceable. We develop an iterative procedure to extend… (More)

We consider the domination number of the queens graph Qn and show that if, for some ÿxed k, there is a dominating set of Q 4k+1 of a certain type with cardinality 2k + 1, then for any n large enough, (Qn) 6 [(3k + 5)=(6k + 3)]n + O(1). The same construction shows that for any m ¿ 1 and n = 2(6m − 1)(2k + 1) − 1, (Q t n) 6 [(2k + 3)=(4k + 2)]n + O(1), where… (More)

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey… (More)

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P. We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r 4, r-regular graphs with girth at… (More)

The relationship ρ L (G) ≤ ρ(G) ≤ γ (G) between the lower packing number ρ L (G), the packing number ρ(G) and the domination number γ (G) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers… (More)