EKR-Type Theorems for Pendant Graph Constructions

TL;DR

This paper investigates Erd ext{"o}s--Ko--Rado (EKR) properties for independent sets in pendant graph constructions, providing new combinatorial proofs and extending known results to generalized pendant complete graphs and paths.

Contribution

It offers a new combinatorial proof that pendant complete graphs are r-EKR for n ≥ 2r, extends this to generalized pendant complete graphs, and analyzes EKR properties of pendant paths.

Findings

Pendant complete graphs are r-EKR for n ≥ 2r.

Generalized pendant complete graphs are r-EKR for n ≥ 2r.

Pendant paths are not (n-k)-EKR for certain n and k.

Abstract

The classical Erd\H{o}s--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.