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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)
No part of this work may be reproduced by print, photocopy or any other means without the permission in writing from the author. Preface In the name of Allaah, the Most Gracious and the Most Merciful. All the praises and thanks are to Allaah, the Lord of the 'aalamiin (mankind, jinn and all that exists). in three topics of graph theory, namely: spanning(More)