On the thickness of graphs of given degree

Abstract

The results presented here refer to the determination of the thickness of a graph, that is, the minimum number of planar subgraphs into which the graph can be decomposed. A useful general, preliminary result obtained here is Theorem 8: a planar graph has a planar representation for any arbitrary placement of its nodes in the plane. It is then proved that if we have positive integers D and T, such that any graph of degree at most D has thickness at most T, the following hold. Theorem 9: Any graph of degree d has thickness at most T roof{ (d + 1) D } . Theorem 10: Any graph of degree d can always be embedded in a regular graph G° of any degree ƒ ⩾ d. Corollary 5: Any graph of degree d has thickness at most roof( d 2 ). Theorem 12: With D and T defined as above, we have D ⩽ 4 T − 2; Corollary 6: If T = 2, then D ⩽ 6. We further conjecture that, indeed, the thickness of any graph of degree not exceeding 6 is never more than 2; and that, more generally, T = roof{ (D + 2) 4 )} . Since the design and fabrication of VLSI computer chips is essentially a concrete representation of the planar decomposition of a graph, all these results are of direct practical interest.