**Commenced**in January 2007

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**Paper Count:**30174

##### A Neighborhood Condition for Fractional k-deleted Graphs

**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.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1331639

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