Graph Universal Cycles: Compression and Connections to Universal Cycles

TL;DR

This paper introduces graph universal partial cycles, a compact representation for various graph classes, and explores their connections to universal cycles, including new constructions and existence proofs for different graph types.

Contribution

It presents the concept of graph universal partial cycles, establishes their relationships with existing universal cycles, and proves their existence for multiple graph classes including labeled, threshold, and permutation graphs.

Findings

Graph universal partial cycles provide a more compact representation of graph classes.

Existence of graph universal cycles and partial cycles is proven for several graph types.

New constructions and overlap forms of universal cycles are introduced for permutation graphs.

Abstract

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal cycles introduced in 2010. We introduce graph universal partial cycles, a more compact representation of graph classes, which use "do not know" edges. We show how to construct graph universal partial cycles for labeled graphs, threshold graphs, and permutation graphs. For threshold graphs and permutation graphs, we demonstrate that the graph universal cycles and graph universal partial cycles are closely related to universal cycles and compressed universal cycles, respectively. Using the same connection, for permutation graphs, we define and prove the existence of an $s$-overlap form of graph universal cycles. We also prove the existence of a generalized form of graph universal cycles for unlabeled graphs.