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Let α ∈ (0, 1) and let G = (V G , E G) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set D ⊆ V G is called an α-dominating set of G, if |N G (u) ∩ D| ≥ αd G (u) for all u ∈ V G \ D. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

All values of n for which there exist a selfcomplementary circulant graph of order n are determined. Let G be a finite , simple , undirected graph. The graph G is selfcompletementary if G is isomorphic to its complement G ; in symbols , G Ӎ G . It is well known that a selfcomplementary graph of order n exists if f n ϵ 0 or 1 (mod 4). Moreover , a regular… (More)

It was proved by Fronček, Jerebic, Klavžar, and Kovář that if a complete bi-partite graph K n,n with a perfect matching removed can be covered by k bicliques, then n ≤ k k 2. We give a slightly simplified proof and we show that the result is tight. Moreover we use the result to prove analogous bounds for coverings of some other classes of graphs by… (More)

We examine decompositions of complete graphs with an even number of vertices, K 2n , into n isomorphic spanning trees. While methods of such decompositions into symmetric trees have been known, we develop here a more general method based on a new type of vertex labelling, called flexible q-labelling. This labelling is a generalization of labellings… (More)

The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's Theorem, the strong isometric dimension of the Hamming graphs K 2 K n is determined.

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σ k > n+k 2 −4k+7 (where k is an arbitrary constant), then G has a 2-factor with at most k − 1 components. As a second main result, we present classes of graphs C 1 ,. .. , C 8 such that every sufficiently large connected claw-free graph satisfying degree condition σ 6… (More)

Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection ℓ : V → {1, 2,. .. , n} for which there exists a positive integer k such that ∑ x∈N (v) ℓ(x) = k for all v ∈ V , where N (v) is the open neighborhood of v. In this paper we deal with circulant graphs C(1, p). The circulant graph C n (1, p) is the graph on the vertex set n −… (More)