A Note on the Immersion Number of Generalized Mycielski Graphs

TL;DR

This paper investigates the immersion number of generalized Mycielski graphs, providing lower bounds based on the original graph's immersion number and introducing the 'distinct neighbor property' to analyze these bounds.

Contribution

It introduces the 'distinct neighbor property' for immersions and establishes lower bounds for the immersion number of generalized Mycielski graphs based on the original graph.

Findings

Lower bounds for im(μ_m(G)) based on im(G)

Examples where im(μ_m(G)) exceeds the lower bound

A conjecture about im(μ_m(K_t))

Abstract

The immersion number of a graph $G$, denoted im$(G)$, is the largest $t$ such that $G$ has a $K_t$-immersion. In this note we are interested in determining the immersion number of the $m$-Mycielskian of $G$, denoted $\mu_m(G)$. Given the immersion number of $G$ we provide a lower bound for im$(\mu_m(G))$. To do this we introduce the "distinct neighbor property" of immersions. We also include examples of classes of graphs where im$(\mu_m(G))$ exceeds the lower bound. We conclude with a conjecture about im$(\mu_m(K_t))$.