Uniformly resolvable decompositions of K_v into paths on two, three and four vertices

TL;DR

This paper determines the spectrum of uniformly resolvable decompositions of complete graphs into paths of lengths 2, 3, and 4, providing a complete characterization of such decompositions.

Contribution

It completely characterizes the spectrum of uniformly resolvable decompositions of K_v into P_2, P_3, and P_4, filling a gap in combinatorial design theory.

Findings

Spectrum determined for decompositions into P_2, P_3, and P_4

Complete classification of such decompositions

Advances understanding of resolvable graph decompositions

Abstract

In this paper we consider uniformly resolvable decompositions of the complete graph K_v into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either P_2, P_3 and P_4.