The Extension Degree Conditions for Fractional Factor

In Gao’s previous work, the authors determined several degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if b = f ( x ) = g ( x ) = a for all vertices x in G . In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference Δ between g ( x ) and f ( x ) for every vertex x in G . These obtained new degree conditions reformulate Gao’s previous conclusions, and show how Δ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting f ( x ) = g ( x ) + Δ.

[1]  Sizhong Zhou,et al.  Some results about component factors in graphs , 2019, RAIRO Oper. Res..

[2]  Si-zhong Zhou,et al.  Some Existence Theorems on All Fractional (g, f)-factors with Prescribed Properties , 2014 .

[3]  Yaojun Chen,et al.  Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs , 2018, Open Physics.

[4]  Zhiren Sun,et al.  A result on r-orthogonal factorizations in digraphs , 2017, Eur. J. Comb..

[5]  Chunxiang Wang,et al.  Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs , 2018, Applied Mathematics and Nonlinear Sciences.

[6]  Sizhong Zhou,et al.  Two sufficient conditions for the existence of path factors in graphs , 2018, Scientia Iranica.

[7]  Wei Gao,et al.  An independent set degree condition for fractional critical deleted graphs , 2019, Discrete & Continuous Dynamical Systems - S.

[8]  Si-zhong Zhou,et al.  Neighborhood Conditions for Fractional ID-k-factor-critical Graphs , 2018, Acta Mathematicae Applicatae Sinica, English Series.

[9]  Juan Luis García Guirao,et al.  Two Tight Independent Set Conditions for Fractional (g, f, m)-Deleted Graphs Systems , 2018 .

[10]  Sizhong Zhou,et al.  Remarks on Fractional ID-k-factor-critical Graphs , 2019, Acta Mathematicae Applicatae Sinica, English Series.

[11]  Li Liang,et al.  Degree Conditions for Fractional $$(g,f,n',m)$$(g,f,n′,m)-Critical Deleted Graphs and Fractional ID-(g, f, m)-Deleted Graphs , 2016 .

[12]  Aleksandra Tepeh,et al.  Convexity result and trees with large Balaban index , 2018 .

[13]  Sizhong Zhou,et al.  An existence theorem on fractional deleted graphs , 2015, Period. Math. Hung..

[14]  Sizhong Zhou,et al.  Remarks on orthogonal factorizations of digraphs , 2014, Int. J. Comput. Math..

[15]  Yang Xu,et al.  A theorem on fractional ID-(g, f)-factor-critical graphs , 2015, Contributions Discret. Math..

[16]  D. Dimitrov,et al.  Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs , 2019, Discrete & Continuous Dynamical Systems - S.

[17]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.