Improved book-embeddings of incomplete hypercubes

Abstract

In this paper, we show that any incomplete hypercube with, at most, 2 n + 2 n − 1 + 2 n − 2 vertices can be embedded in n − 1 pages for all n ≥ 4 . For the case n ≥ 4 , this result improves Fang and Lai’s result that any incomplete hypercube with, at most, 2 n + 2 n − 1 vertices can be embedded in n − 1 pages for all n ≥ 2 . Besides this, we show that the result can be further improved when n is large — e.g., any incomplete hypercube with at most 2 n + 2 n − 1 + 2 n − 2 + 2 n − 7 (respectively, 2 n + 2 n − 1 + 2 n − 2 + 2 n − 7 + 2 n − 230 ) vertices can be embedded in n − 1 pages for all n ≥ 9 (respectively, n ≥ 232 ).