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A note on the independent domination number of subset graph

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Abstract

The independent domination number i(G) (independent number β(G)) is the minimum (maximum) cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree δ ≤ 1/2n satisfies i(G) ≤ ⌈2n/3δ⌉ 1/2δ. For 1 ≤ k ≤ l ≤ m, the subset graph S m (k, l) is the bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(S m (k, l)) and prove that if k + l = m then Haviland’s conjecture holds for the subset graph S m (k, l). Furthermore, we give the exact value of β(S m (k, l)).

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Authors and Affiliations

  1. Dept. of Math., Shantou University, Shantou, Guangdong, 515063, P.R. China

    Xue-gang Chen

  2. The College of Information Science and Engineering, Shandong University of Science and Technology, Taian, Shandong Province, 271019, P.R. China

    De-xiang Ma

  3. Dept. of Mathematics, Langfang Normal College, Langfang, Hebei, 065000, P.R. China

    Hua-Ming Xing

  4. Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China

    Liang Sun

Authors
  1. Xue-gang Chen
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  2. De-xiang Ma
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  3. Hua-Ming Xing
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  4. Liang Sun
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Corresponding author

Correspondence to Xue-gang Chen.

Additional information

This work was supported by National Natural Sciences Foundation of China (19871036).

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Chen, Xg., Ma, Dx., Xing, HM. et al. A note on the independent domination number of subset graph. Czech Math J 55, 511–517 (2005). https://doi.org/10.1007/s10587-005-0042-9

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  • DOI: https://doi.org/10.1007/s10587-005-0042-9

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