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The generalized 3-connectivity of the Mycielskian of a graph

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  • Li, Shasha
  • Zhao, Yan
  • Li, Fengwei
  • Gu, Ruijuan

Abstract

The generalized k-connectivity κk(G) of a graph G is a generalization of the concept of the traditional connectivity, which can serve for measuring the capability of a graph G to connect any k vertices in G. In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. In this paper, we investigate the relation between the generalized 3-connectivity of the Mycielskian of a graph G and the generalized 3-connectivity of G, and show that κ3(μ(G))≥κ3(G)+1. Moreover, by this result, we completely determine the generalized 3-connectivity of the Mycielskian of the tree Tn, the complete graph Kn and the complete bipartite graph Ka,b.

Suggested Citation

  • Li, Shasha & Zhao, Yan & Li, Fengwei & Gu, Ruijuan, 2019. "The generalized 3-connectivity of the Mycielskian of a graph," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 882-890.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:882-890
    DOI: 10.1016/j.amc.2018.11.006
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    References listed on IDEAS

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    1. Li, Shasha & Tu, Jianhua & Yu, Chenyan, 2016. "The generalized 3-connectivity of star graphs and bubble-sort graphs," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 41-46.
    2. Hui Lei & Shasha Li & Henry Liu & Yongtang Shi, 2018. "Rainbow vertex connection of digraphs," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 86-107, January.
    3. Yongtang Shi & Meiqin Wei & Jun Yue & Yan Zhao, 2017. "Coupon coloring of some special graphs," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 156-164, January.
    4. Shasha Li & Xueliang Li, 2012. "Note on the hardness of generalized connectivity," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 389-396, October.
    5. Li, Hengzhe & Ma, Yingbin & Yang, Weihua & Wang, Yifei, 2017. "The generalized 3-connectivity of graph products," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 77-83.
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    2. Zhu, Wen-Han & Hao, Rong-Xia & Feng, Yan-Quan & Lee, Jaeun, 2023. "The 3-path-connectivity of the k-ary n-cube," Applied Mathematics and Computation, Elsevier, vol. 436(C).

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