Cliques in realization graphs

TL;DR

This paper characterizes the structure of realization graphs of degree sequences to determine when they contain large cliques or are complete, extending previous work on triangle-free cases.

Contribution

It provides a structural description of realizations that exactly determine the presence of large cliques in realization graphs, and characterizes degree sequences for complete realization graphs.

Findings

Identifies structures in realizations that determine clique sizes.

Characterizes degree sequences with complete realization graphs.

Extends previous triangle-free characterization to larger cliques.

Abstract

The realization graph $\mathcal{G}(d)$ of a degree sequence $d$ is the graph whose vertices are labeled realizations of $d$, where edges join realizations that differ by swapping a single pair of edges. Barrus [On realization graphs of degree sequences, Discrete Mathematics, vol. 339 (2016), no. 8, pp. 2146-2152] characterized $d$ for which $\mathcal{G}(d)$ is triangle-free. Here, for any $n \geq 4$, we describe a structure in realizations of $d$ that exactly determines whether $G(d)$ has a clique of size $n$. As a consequence we determine the degree sequences $d$ for which $\mathcal{G}(d)$ is a complete graph on $n$ vertices.