The metamorphosis of lambda-fold 4-wheel systems into lambda-fold 4-cycle systems

Abstract

A 4-wheel is a simple graph on 5 vertices with 8 edges, formed by taking a 4-cycle and joining a fifth vertex (the centre of the 4-wheel) to each of the other four vertices. A λ-fold 4-wheel system of order n is an edge-disjoint decomposition of the complete multigraph λKn into 4-wheels. Here, with five isolated possible exceptions when λ = 2, we give necessary and sufficient conditions for a λ-fold 4-wheel system of order n to be transformed into a λ-fold 4-cycle system of order n by removing the centre vertex from each 4-wheel, and its four adjacent edges (retaining the 4-cycle wheel "rim"), and reassembling these edges adjacent to wheel centres into 4-cycles.