$ Z_{3} $-connectivity of graphs with independence number at most 3

Chunyan Qin; Xiaoxia Zhang; Liangchen Li; Chunyan Qin; Xiaoxia Zhang; Liangchen Li

doi:10.3934/math.2023164

AIMS Mathematics

2023, Volume 8, Issue 2: 3204-3209. doi: 10.3934/math.2023164

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$ Z_{3} $-connectivity of graphs with independence number at most 3

1.
Department of Mathematics, Luoyang Normal University, Luoyang, China
2.
Department of Mathematics, Xinyang Normal University, Xinyang, China

Received: 28 August 2022 Revised: 01 November 2022 Accepted: 08 November 2022 Published: 16 November 2022
MSC : 05C21

It was conjectured by Jaeger et al. that all 5-edge-connected graphs are $ Z_3 $-connected. In this paper, we confirm this conjecture for all 5-edge-connected graphs with independence number at most 3.
- nowhere-zero flows,
- $ Z_{3} $-connectivity,
- independence number
Citation: Chunyan Qin, Xiaoxia Zhang, Liangchen Li. $ Z_{3} $-connectivity of graphs with independence number at most 3[J]. AIMS Mathematics, 2023, 8(2): 3204-3209. doi: 10.3934/math.2023164

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Abstract

It was conjectured by Jaeger et al. that all 5-edge-connected graphs are $ Z_3 $-connected. In this paper, we confirm this conjecture for all 5-edge-connected graphs with independence number at most 3.

References

[1]	B. A. Bondy, U. S. R. Murty, Graph Theory, London: Springer, 2008.
[2]	J. Chen, E. M. Eschen, H. J. Lai, Group connectivity of certain graphs, Ars Combinatoria, 89 (2008), 141–158.
[3]	M. DeVos, R. Xu, G. Yu, Nowhere-zero $Z_3$-flows through $Z_3$-connectivity, Discrete Math., 306 (2006), 26–30. https://doi.org/10.1016/j.disc.2005.10.019 doi: 10.1016/j.disc.2005.10.019
[4]	F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs-a nonhomogeneous analogue of nowhere-zero flow properties, J. Comb. Theory B, 56 (1992), 165–182. https://doi.org/10.1016/0095-8956(92)90016-Q doi: 10.1016/0095-8956(92)90016-Q
[5]	H. J. Lai, Group connectivity of 3-edge-connected chordal graphs, Graph. Combinator., 16 (2000), 165–176. https://doi.org/10.1007/PL00021177 doi: 10.1007/PL00021177
[6]	J. Li, R. Luo, Y. Wang, Nowhere-zero 3-flow of graphs with small independence number, Discrete Math., 341 (2018), 42–50. https://doi.org/10.1016/j.disc.2017.06.022 doi: 10.1016/j.disc.2017.06.022
[7]	L. M. Lovász, C. Thomassen, Y. Wu, C. Q. Zhang, Nowhere-zero 3-flows and modulo k-orientations, J. Comb. Theory B, 103 (2013), 587–598. https://doi.org/10.1016/j.jctb.2013.06.003 doi: 10.1016/j.jctb.2013.06.003
[8]	R. Luo, Z. Miao, R. Xu, Nowhere-zero 3-flows of graphs with independence number two, Graph. Combinator., 29 (2013), 1899–1907. https://doi.org/10.1007/s00373-012-1238-z doi: 10.1007/s00373-012-1238-z
[9]	R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and $Z_3$-connectivity, Eur. J. Comb., 29 (2008), 1587–1595. https://doi.org/10.1016/j.ejc.2007.11.014 doi: 10.1016/j.ejc.2007.11.014
[10]	W. T. Tutte, A contribution to the theory of chromatic polynomials, Can. J. Math., 6 (1954), 80–91. https://doi.org/10.4153/CJM-1954-010-9 doi: 10.4153/CJM-1954-010-9
[11]	W. T. Tutte, On the algebraic theory of graph colorings, J. Comb. Theory, 1 (1966), 15–50. https://doi.org/10.1016/S0021-9800(66)80004-2 doi: 10.1016/S0021-9800(66)80004-2
[12]	F. Yang, X. Li, L. Li, ${Z}_3$-connectivity with independent number 2, Graph. Combinator., 32 (2016), 419–429. https://doi.org/10.1007/s00373-015-1556-z doi: 10.1007/s00373-015-1556-z

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