On restricted edge-connectivity of replacement product graphs
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This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, we construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d ≥ 5 and 1 ≤ s ≤ d −3, which answers a problem proposed ten years ago.
Keywordsgraph theory connectivity restricted edge-connectivity replacement product Cayley graph
MSC(2010)05C40 68M15 68R10
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This work was supported by National Natural Science Foundation of China (Grant Nos. 61272008 and 11571044), University Natural Science Research Project of Anhui Province (Grant No. KJ2016A003) and Scientific Research Fund of Anhui University of Finance & Economics (Grant No. ACKY1532).
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