Abstract
A graph G is called a fractional [a, b]-covered graph if for each e ∈ E(G), G contains a fractional [a, b]-factor covering e. A graph G is called a fractional (a, b, k)-critical covered graph if for any W ⊆ V(G) with |W| = k, G − W is fractional [a, b]-covered, which was first defined and investigated by Zhou, Xu and Sun [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (a, b, k)-critical covered graphs, Information Processing Letters 152(2019)105838]. In this work, we proceed to study fractional (a, b, k)-critical covered graphs and derive a result on fractional (a, b, k)-critical covered graphs depending on minimum degree and neighborhoods of independent sets.
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Change history
13 August 2022
The word “Technology” in the second affiliation of Zhang Wei was misspelled as “Technlogogy”.
References
Correa, J., Matamala, M. Some remarks about factors of graphs. Journal of Graph Theory, 57: 265–274 (2008)
Gao, W., Wang, W., Dimitrov, D. Toughness condition for a graph to be all fractional (g, f, n)-critical deleted. Filomat, 33: 2735–2746 (2019)
Gao, W., Wang, W., Guirao, J. The extension degree conditions for fractional factor. Acta Mathematica Sinica-English Series, 36: 305–317 (2020)
Iida, T., Nishimura, T. Neighborhood conditions and k-factors. Tokyo Journal of Mathematics, 20: 411–418 (1997)
Kano, M., Tokushige, N. Binding numbers and f-factors of graphs. Journal of Combinatorial Theory, Series B, 54: 213–221 (1992)
Katerinis, P. Fractional l-factors in regular graphs. Australasian Journal of Combinatorics, 73: 432–439 (2019)
Kimura, K. f-factors, complete-factors, and component-deleted subgraphs. Discrete Mathematics, 313: 1452–1463 (2013)
Li, Z., Yan, G., Zhang, X. On fractional (g, f)-covered graphs. OR Transactions (China), 6: 65–68 (2002)
Liu, G., Zhang, L. Toughness and the existence of fractional k-factors of graphs. Discrete Mathematics, 308: 1741–1748 (2008)
Lu, H. Simplified existence theorems on all fractional [a, b]-factors. Discrete Applied Mathematics, 161: 2075–2078 (2013)
Lu, H., Yu, Q. General fractional f-factor numbers of graphs. Applied Mathematics Letters, 24: 519–523 (2011)
Lv, X. A degree condition for fractional (g, f, n)-critical covered graphs. AIMS Mathematics, 5: 872–878 (2020)
Nam, Y. Binding numbers and connected factors. Graphs and Combinatorics, 26: 805–813 (2010)
Wang, S., Zhang, W. On k-orthogonal factorizations in networks. RAIRO-Operations Research, 55: 969–977 (2021)
Wang, S., Zhang, W. Research on fractional critical covered graphs. Problems of Information Transmission, 56: 270–277 (2020)
Wang, S., Zhang, W. Toughness for fractional (2, b, k)-critical covered graphs. Journal of the Operations Research Society of China, DOI: https://doi.org/10.1007/s40305-021-00359-4
Woodall, D. R. k-factors and neighborhoods of independent sets in graphs. Journal of the London Mathematical Society, 41: 385–392 (1990)
Yuan, Y., Hao, R. A degree condition for fractional [a, b]-covered graphs. Information Processing Letters, 143: 20–23 (2019)
Yuan, Y., Hao, R. A neighborhood union condition for fractional ID-[a, b]-factor-critical graphs. Acta Mathematicae Applicatae Sinica-English Series, 34: 775–781 (2018)
Yuan, Y., Hao, R. Independence number, connectivity and all fractional (a, b, k)-critical graphs. Discussiones Mathematicae Graph Theory, 39: 183–190 (2019)
Zhou, S. A neighborhood union condition for fractional (a, b, k)-critical covered graphs. Discrete Applied Mathematics, DOI: https://doi.org/10.1016/j.dam.2021.05.022
Zhou, S. A result on fractional (a, b, k)-critical covered graphs. Acta Mathematicae Applicatae Sinica-English Series, 37: 657–664 (2021)
Zhou, S. Binding numbers and restricted fractional (g, f)-factors in graphs. Discrete Applied Mathematics, 305: 350–356 (2021)
Zhou, S. Remarks on path factors in graphs. RAIRO-Operations Research, 54: 1827–1834 (2020)
Zhou, S., Bian, Q., Pan, Q. Path factors in subgraphs. Discrete Applied Mathematics, DOI: https://doi.org/10.1016/j.dam.2021.04.012
Zhou, S., Bian, Q., Sun, Z. Two sufficient conditions for component factors in graphs. Discussiones Mathematicae Graph Theory, DOI: https://doi.org/10.7151/dmgt.2401
Zhou, S., Liu, H. Discussions on orthogonal factorizations in digraphs. Acta Mathematicae Applicatae Sinica-English Series, DOI: https://doi.org/10.1007/s10255-022-1086-4
Zhou, S., Liu, H., Xu, Y. A note on fractional ID-[a, b]-factor-critical covered graphs. Discrete Applied Mathematics, DOI: https://doi.org/10.1016/j.dam.2021.03.004
Zhou, S., Sun, Z., Liu, H. On P≥3-factor deleted graphs. Acta Mathematicae Applicatae Sinica-English Series, 38: 178–186 (2022)
Zhou, S., Sun, Z., Pan, Q. A sufficient condition for the existence of restricted fractional (g, f)-factors in graphs. Problems of Information Transmission, 56: 332–344 (2020)
Zhou, S., Wu, J., Bian, Q. On path-factor critical deleted (or covered) graphs. Aequationes Mathematicae, DOI: https://doi.org/10.1007/s00010-021-00852-4
Zhou, S., Wu, J., Xu, Y. Toughness, isolated toughness and path factors in graphs. Bulletin of the Australian Mathematical Society, DOI: https://doi.org/10.1017/S0004972721000952
Zhou, S., Xu, Y., Sun, Z. Degree conditions for fractional (a, b, k)-critical covered graphs. Information Processing Letters, 152: 105838 (2019)
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The authors are very grateful to the anonymous referees for their valuable comments and suggestions, which have greatly improved the final version of this article.
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Zhang, W., Wang, Sf. Discussion on Fractional (a, b, k)-critical Covered Graphs. Acta Math. Appl. Sin. Engl. Ser. 38, 304–311 (2022). https://doi.org/10.1007/s10255-022-1076-6
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DOI: https://doi.org/10.1007/s10255-022-1076-6