A large hole in pseudo-random graphs

TL;DR

Problem: determine the existence and size of induced holes in pseudo-random graphs, i.e., induced cycles in $(n,d,\lambda)$-graphs with small spectral ratio $\lambda/d$, relating cycle length to density via $\delta_2 n/d$. Approach: develop a percolation-based method by studying site-percolated subgraphs $G_p$ and a DFS-guided exploration to guarantee a long induced path in $G_p$, which is then extended to a long induced cycle in $G$ using a neighborhood-expansion argument; spectral tools control edge distributions and the size of induced substructures. Contributions: (i) existence of an induced cycle of length at least $\delta_2 n/d$ (tight up to constants), (ii) exponential lower bound on $\mu(G)$, $\mu(G)\ge \exp\left(\delta_2 n \log d / d\right)$, with matching upper-bound constructions giving tightness up to constants in the exponent, and (iii) lexicographic-product constructions yielding near-optimal upper bounds on $\mu(G)$ and a randomized linear-time algorithm to find long holes. Significance: extends results from random graphs to pseudo-random expanders, clarifies the role of spectral ratio, and connects percolation techniques to cycle lengths and subgraph diversity with algorithmic implications.

Abstract

We show that there exist constants $δ_1,δ_2>0$ such that if $G$ is an $(n,d,λ)$-graph with $λ/d\leδ_1$, then $G$ contains an induced cycle of length at least $δ_2n/d$. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such $G$ is at least exponential in $n\log d/d$, and further demonstrate that this is tight up to a constant factor in the exponent.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Lemma 2.4 in zbMATH07751060, see also zbMATH06534995
• Theorem 2.4: Theorem 2 of zbMATH07751060
• Theorem 2.5: Theorem 4 of zbMATH07751060