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Let G be a 3-connected bipartite graph with partite sets X ∪ Y which is em-beddable in the torus. We shall prove that G has a Hamiltonian cycle if (i) G is blanced , i.e., |X| = |Y |, and (ii) each vertex x ∈ X has degree four. In order to prove the result, we establish a result on orientations of quadrangular torus maps possibly with multiple edges. This(More)
A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The weight of a graph is defined as the sum of the weights of its edges. In 2-edge-colored complete graph, by using Ramsey-type theorems, we obtain the existence of monochromatic subgraph which have many edges compared with its order. In this paper, we(More)
For a positive integer k, a set of vertices S in a graph G is said to be a k-dominating set if each vertex x in V (G) − S has at least k neighbors in S. The order of a smallest k-dominating set of G is called the k-domination number of G and is denoted by γ k (G). In Blidia, Chellali and Favaron [Australas. they proved that a tree T satisfies α(T) ≤ γ 2 (T)(More)
Ota and Tokuda [2] gave a minimum degree condition for a K1,n-free graph to have a 2-factor. Though their condition is best-possible, their sharpness examples have edge-connectivity one. In this paper, we improve their minimum degree condition for K1,n-free graphs with large connectivity or large edge-connectivity. Our bound is sharp, and together with Ota(More)
In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. For d ≥ 2, let G d be the set of connected graphs of minimum degree at least d. Let F 1 and F 2 be connected graphs and let H be a set of connected graphs. Then {F 1 , F 2 } is said to be a forbidden pair for H if every {F 1 , F 2 }-free graph in H of(More)
A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree d w (v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted(More)