# A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs

• Volume: 36, Issue: 2, page 409-418
• ISSN: 2083-5892

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## Abstract

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Let G be a graph of order n, and let a and b be two integers with 1 ≤ a ≤ b. Let h : E(G) → [0, 1] be a function. If a ≤ ∑e∋x h(e) ≤ b holds for any x ∈ V (G), then we call G[Fh] a fractional [a, b]-factor of G with indicator function h, where Fh = {e ∈ E(G) : h(e) > 0}. A graph G is fractional independent-set-deletable [a, b]-factor-critical (in short, fractional ID-[a, b]- factor-critical) if G − I has a fractional [a, b]-factor for every independent set I of G. In this paper, it is proved that if [...] for any two nonadjacent vertices x, y ∈ V (G), then G is fractional ID-[a, b]-factor-critical. Furthermore, it is shown that this result is best possible in some sense.

## How to cite

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Sizhong Zhou, Fan Yang, and Zhiren Sun. "A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs." Discussiones Mathematicae Graph Theory 36.2 (2016): 409-418. <http://eudml.org/doc/277129>.

@article{SizhongZhou2016,
abstract = {Let G be a graph of order n, and let a and b be two integers with 1 ≤ a ≤ b. Let h : E(G) → [0, 1] be a function. If a ≤ ∑e∋x h(e) ≤ b holds for any x ∈ V (G), then we call G[Fh] a fractional [a, b]-factor of G with indicator function h, where Fh = \{e ∈ E(G) : h(e) > 0\}. A graph G is fractional independent-set-deletable [a, b]-factor-critical (in short, fractional ID-[a, b]- factor-critical) if G − I has a fractional [a, b]-factor for every independent set I of G. In this paper, it is proved that if [...] for any two nonadjacent vertices x, y ∈ V (G), then G is fractional ID-[a, b]-factor-critical. Furthermore, it is shown that this result is best possible in some sense.},
author = {Sizhong Zhou, Fan Yang, Zhiren Sun},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; minimum degree; neighborhood; fractional [a; b]-factor; fractional ID-[a; b]-factor-critical graph; fractional -factor; fractional $ID-[a, b]$-factor-critical graph},
language = {eng},
number = {2},
pages = {409-418},
title = {A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs},
url = {http://eudml.org/doc/277129},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Sizhong Zhou
AU - Fan Yang
AU - Zhiren Sun
TI - A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2016
VL - 36
IS - 2
SP - 409
EP - 418
AB - Let G be a graph of order n, and let a and b be two integers with 1 ≤ a ≤ b. Let h : E(G) → [0, 1] be a function. If a ≤ ∑e∋x h(e) ≤ b holds for any x ∈ V (G), then we call G[Fh] a fractional [a, b]-factor of G with indicator function h, where Fh = {e ∈ E(G) : h(e) > 0}. A graph G is fractional independent-set-deletable [a, b]-factor-critical (in short, fractional ID-[a, b]- factor-critical) if G − I has a fractional [a, b]-factor for every independent set I of G. In this paper, it is proved that if [...] for any two nonadjacent vertices x, y ∈ V (G), then G is fractional ID-[a, b]-factor-critical. Furthermore, it is shown that this result is best possible in some sense.
LA - eng
KW - graph; minimum degree; neighborhood; fractional [a; b]-factor; fractional ID-[a; b]-factor-critical graph; fractional -factor; fractional $ID-[a, b]$-factor-critical graph
UR - http://eudml.org/doc/277129
ER -

## References

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1. [1] S. Zhou, Z. Sun and H. Liu, A minimum degree condition for fractional ID-[a, b]- factor-critical graphs, Bull. Aust. Math. Soc. 86 (2012) 177-183. doi:10.1017/S0004972711003467[Crossref][WoS] Zbl1252.05182
2. [2] H. Matsuda, A neighborhood condition for graphs to have [a, b]-factors, Discrete Math. 224 (2000) 289-292. doi:10.1016/S0012-365X(00)00140-0[Crossref] Zbl0966.05062
3. [3] J. Ekstein, P. Holub, T. Kaiser, L. Xiong and S. Zhang, Star subdivisions and con- nected even factors in the square of a graph, Discrete Math. 312 (2012) 2574-2578. doi:10.1016/j.disc.2011.09.004[Crossref][WoS] Zbl1246.05095
4. [4] H. Lu, Simplified existence theorems on all fractional [a, b]-factors, Discrete Appl. Math. 161 (2013) 2075-2078. doi:10.1016/j.dam.2013.02.006[Crossref]
5. [5] S. Zhou, Independence number, connectivity and (a, b, k)-critical graphs, Discrete Math. 309 (2009) 4144-4148. doi:10.1016/j.disc.2008.12.013[Crossref][WoS]
6. [6] S. Zhou, A sufficient condition for graphs to be fractional (k,m)-deleted graphs, Appl. Math. Lett. 24 (2011) 1533-1538. doi:10.1016/j.aml.2011.03.041[WoS][Crossref]
7. [7] S. Zhou and H. Liu, Neighborhood conditions and fractional k-factors, Bull. Malays. Math. Sci. Soc. 32 (2009) 37-45. Zbl1209.05196
8. [8] K. Kimura, f-factors, complete-factors, and component-deleted subgraphs, Discrete Math. 313 (2013) 1452-1463. doi:10.1016/j.disc.2013.03.009[Crossref][WoS] Zbl1279.05056
9. [9] M. Kouider and Z. Lonc, Stability number and [a, b]-factors in graphs, J. Graph Theory 46 (2004) 254-264. doi:10.1002/jgt.20008[Crossref]
10. [10] R. Chang, G. Liu and Y. Zhu, Degree conditions of fractional ID-k-factor-critical graphs, Bull. Malays. Math. Sci. Soc. 33 (2010) 355-360. Zbl1260.05119
11. [11] S. Zhou, Q. Bian and J. Wu, A result on fractional ID-k-factor-critical graphs, J. Combin. Math. Combin. Comput. 87 (2013) 229-236. Zbl1293.05310
12. [12] S. Zhou, Binding numbers for fractional ID-k-factor-critical graphs, Acta Math. Sin. Engl. Ser. 30 (2014) 181-186. doi:10.1007/s10114-013-1396-9[Crossref][WoS] Zbl1287.05124
13. [13] G. Liu and L. Zhang, Fractional (g, f)-factors of graphs, Acta Math. Sci. Ser. B 21 (2001) 541-545. Zbl0989.05086

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