On the Equivalences of the Wheel $W_5$ the Prism $P$ and the Bipyramid $B_5$

Abstract

The main results of this paper are the discovery of infinite families of flow equivalent pairs of $B_5$ and $W_5$, amalamorphs, and infinite families of chromatically equivalent pairs of $P$ and $W_5^{\ast }$; homeomorphs, where $B_5$ is $K_5$ with one edge deleted, $P$ is the Prism graph, and $W_5$ is the join of $K_1$ and a cycle on 4 vertices. Six families of $B_5$ amalamorphs, with two families having 6 parameters, and 9 families of $W_5$ amalamorphs, with one family having 4 parameters, are discovered. Since $B_5$ and $W_5$ are both planar, all these results obtained can be stated in terms of chromatically equivalent pairs of $B_5^{\ast }$ and $W_5^{\ast }$ homeomorphs. Also, three conjectures are made about the non-existence of flow-equivalent amalamorphs or chromatically equivalent homeomorphs of certain graphs.