Unconditional Lower Bounds for Degree Fault Tolerant Spanners

2026-07-08

AI Summary

This study addresses the problem of determining the minimum size of fault-tolerant graph spanners that maintain a multiplicative stretch of \( (2k-1) \) under up to \( f \) edge failures. By introducing a structural analysis based on Wenger graphs and leveraging recent reinterpretations by Szabó and Conlon, the work establishes—unconditionally—the first lower bound of \( \Omega(f^{1-1/k} n^{1+1/k}) \) for \( f \)-edge-fault-tolerant \( (2k-1) \)-spanners, without relying on the Erdős girth conjecture. This bound matches the best-known upper bound up to an \( \exp(k) \) factor, thereby yielding a nearly tight characterization of the optimal spanner size and significantly advancing the foundational understanding of fault-tolerant spanners in graph theory.

Abstract

We study multiplicative graph spanners in the $f$-degree fault tolerant ($f$-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree $f$ temporarily "fails" and is removed from the graph. We prove that there are $n$-node lower bound graphs for which any $f$-DFT $(2k-1)$-stretch spanner $H$ must have size $$|E(H)| \ge Ω\left( f^{1-1/k} n^{1+1/k}\right).$$ This matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erdős. It also matches the current upper bounds, up to a factor of $\texttt{exp}(k)$. Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szabó and by Conlon (Am. Math. Monthly 2021).

Problem

• fault tolerant spanners
• degree fault tolerance
• graph spanners
• lower bounds
• Wenger graphs

Innovation

• fault-tolerant spanners
• unconditional lower bounds
• Wenger graphs
• graph sparsification
• multiplicative stretch