Uniformly resolvable $(C_4, K_{1,3})$-designs of order $v$ and index $2$

TL;DR

This paper fully characterizes the spectrum of uniformly resolvable decompositions of the complete graph with index 2 into classes of either 4-cycles or stars with three leaves, providing a comprehensive understanding of such designs.

Contribution

It completely determines the spectrum for uniformly resolvable decompositions of $2K_v$ into $C_4$ or $K_{1,3}$ classes, a novel and comprehensive result.

Findings

Spectrum fully determined for all cases

Decomposition exists for all specified parameters

Provides a complete classification of such designs

Abstract

In this paper we consider the uniformly resolvable decompositions of the complete graph $2K_v$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the cases in which all the resolution classes are either $C_4$ or $K_{1,3}$.