Search results for: Hongxia Liu

Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Let G be a graph of order n, and let k  2 and m  0 be two integers. Let h : E(G)  [0, 1] be a function. If e∋x h(e) = k holds for each x  V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e  E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0 for any e  E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if (G)  k + m + m k+1 , n  4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)}  n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.

Keywords: Graph, degree condition, fractional k-factor, fractional (k, m)-deleted graph.

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Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Abstract–Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k +3- 42(k - 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k + 1 and |NG(x) ∪ NG(y)| ≥ 1 2 (n + k - 2) for each pair of nonadjacent vertices x, y of G.

Keywords: Graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.

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