**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30848

##### On Fractional (k,m)-Deleted Graphs with Constrains Conditions

**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

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1330151

**References:**

[1] S. Zhou, A neighborhood condition for graphs to be fractional (k,m)- deleted graphs, Glasgow Mathematical Journal 52(1)(2010), 33-40.

[2] S. Zhou, Independence number, connectivity and (a, b, k)-critical graphs, Discrete Mathematics 309(12)(2009), 4144-4148.

[3] S. Zhou and Q. Shen, On fractional (f, n)-critical graphs, Information Processing Letters 109(14)(2009), 811-815.

[4] H. Matsuda, Fan-type results for the existence of

[a, b]-factors, Discrete Mathematics 306(2006), 688-693.

[5] J. R. Correa and M. Matamala, Some remarks about factors of graphs, Journal of Graph Theory 57(2008), 265-274.

[6] H. Liu and G. Liu, Binding number and minimum degree for the existence of (g, f, n)-critical graphs, Journal of Applied Mathematics and Computing 29(1-2)(2009), 207-216.

[7] J. Yu and G. Liu, Fractional k-factors of graphs, Chinese Journal of Engineering Mathematics 22(2)(2005), 377-380.

[8] G. Liu and L. Zhang, Toughness and the existence of fractional k-factors of graphs, Discrete Mathematics 308(2008), 1741-1748.

[9] J. Cai and G. Liu, Stability number and fractional f-factors in graphs, Ars Combinatoria 80(2006), 141-146.

[10] S. Zhou, Some results on fractional k-factors, Indian Journal of Pure and Applied Mathematics 40(2)(2009), 113-121.

[11] S. Zhou, A minimum degree condition of fractional (k,m)-deleted graphs, Comptes rendus Mathematique 347 (21-22)(2009), 1223-1226.

[12] T. Niessen, A Fan-type result for regular factors, Ars combinatoria, 46(1997), 277-285.