Abstract. Let G = (V, E) be a graph of order n. A bijection f : V → {1, 2, . . . , n} is called a distance magic labeling of G if there exists a positive integer k such that
We give two results on domination in graphs, including a proof of a conjecture of Favaron, Henning, Mynhart and Puech [2]. Corollary 2 was found by four separate subsets of the authors. We decided to give this joint presentation of our results. We first offer a result about bipartite graphs.Lemma 1 Let G be a bipartite graph with partite sets (X, Y ) whose vertices in Y are of minimum degree at least 3. Then there exists a set A ⊂ X of size at most |X ∪ Y |/4 such that every vertex in Y is adjacent to a vertex in A.Proof: The proof is by induction on |V (G)| + |E(G)|. The smallest graph as described in the lemma is K 1,3 , for which the statement holds. This gives the start of our induction. Let x = |X| and y = |Y |. If there exists a vertex v in Y of degree at least 4, then delete any edge e incident to v. The subset A of G − e guaranteed by the inductive hypothesis is adjacent in G to every vertex in Y as desired. So we may assume that the vertices in Y are all of
It was proved by Fronček, Jerebic, Klavžar, and Kovář that if a complete bipartite 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 bicliques.
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