EKR-Type Theorems for Pendant Graph Constructions
Michael Carrion, Melissa M. Fuentes, Zaphenath Joseph, Alexander Nappo
TL;DR
This work investigates $r$-EKR properties for independent $r$-sets in pendant graph constructions. Using shifting, shadows, and related combinatorial tools, it proves that pendant complete graphs $K_n^*$ are $r$-EKR for $n \ge 2r$ (with strictness when $n>2r$) and extends the result to generalized pendant complete graphs $K_n^{\mathbf{s}}$ via inductive and local-shift arguments. It also provides explicit counterexamples showing that pendant paths $P_n^*$ fail the $r$-EKR property in several regimes, highlighting a nuanced landscape beyond the clique-based pendants. These results align with the Holroyd–Talbot framework by linking $r$-EKR thresholds to independence parameters and offer tools for analyzing more complex pendant constructions, including cycles and graph powers. The findings advance understanding of when stars are extremal for independent sets in pendant-like graphs and suggest directions for future work in broader graph families.
Abstract
The classical Erdős--Ko--Rado (EKR) theorem characterizes the maximum size of intersecting families of $r$-element subsets of an $n$-element set. We study EKR-type questions for independent $r$-sets in \emph{pendant} graph constructions, obtained by attaching to each base vertex a clique of prescribed size. Our contributions are threefold. We give an alternate and purely combinatorial proof (via shifting and shadows) that the pendant complete graph $K_n^{*}$ is $r$-EKR for $n \ge 2r$, and strictly so for $n>2r$, recovering a result of De Silva, Dionne, Dunkelberg, and Harris. We extend this to \emph{generalized pendant complete graphs}, where every base vertex in the clique supports a clique of arbitrary size, proving that that generalized pendant complete graphs are $r$-EKR whenever $n \ge 2r$. For pendant paths $P_n^{*}$, we provide elementary constructions showing that $P_n^{*}$ is not $(n-k)$-EKR when $n \ge 3k+2$ for $k\ge 2$, not $(n-1)$-EKR for $n\ge 6$, and not $n$-EKR for $n\ge 4$. These results fit naturally into the Holroyd--Talbot perspective relating $r$-EKR thresholds to independence parameters and supply tools for further pendant constructions.