Abstract
Suppose K v is the complete undirected graph with v vertices and K 4 − e is the graph obtained from a complete graph K 4 by removing one edge. Let (K 4 − e)-MRC(v) denote a resolvable covering of K v with copies of K 4 − e with the minimum possible number n(v, K 4 − e) of parallel classes. It is readily verified that \({n(v, K_4-e) \geq \lceil 2(v-1)/5 \rceil}\) . In this article, it is proved that there exists a (K 4 − e)-MRC(v) with \({\lceil 2(v-1)/5 \rceil}\) parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 108, 172, 228, 292, 296, 308, 412. In addition, the known results on the existence of maximum resolvable (K 4 − e)-packings are also improved.
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The research was supported by the National Natural Science Foundation of China under Grant No. 60873267 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y6100126 for R. Su, and by the National Natural Science Foundation of China under Grant Nos. 10771051 and 11001182 for L. Wang.
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Su, R., Wang, L. Minimum Resolvable Coverings of K v with Copies of K 4 − e . Graphs and Combinatorics 27, 883–896 (2011). https://doi.org/10.1007/s00373-010-1003-0
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DOI: https://doi.org/10.1007/s00373-010-1003-0