The number of labeled graphs placeable by a given permutation

Abstract

Let S be a finite set and σ a permutation on S. The permutation σ* on the set of 2‐subsets of S is naturally induced by σ. Suppose G is a graph and V(G), E(G) are the vertex set, the edge set, respectively. Let V(G) = S. If E(G) and σ*(E(G)), the image of E(G) by σ*, have no common element, then G is said to be placeable by σ. This notion is generalized as follows. If any two sets of {E(G), (σ1)*(E(G)),…,(σl−1)* (E(G))} have no common element, then G is said to be I‐placeable by σ. In this paper, we count the number of labeled graphs which are I‐placeable by a given permutation. At first, we introduce the interspaced Ith Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced Ith Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, we show the number of labeled graphs which are I‐placeable by a given permutation is a product of the interspaced Ith Lucas numbers. Finally, using a property of the generalized binomial series, we count the number of labeled graphs of size k which are I‐placeable by σ. © 1996 John Wiley & Sons, Inc.