Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k)
Year: 2026
Type: erratum
Abstract: Computational and constructive results on the Erdos-Gyarfas conjecture(every graph of minimum degree 3 contains a cycle whose length is a powerof two). Main results: (1) every graph with minimum degree at least 3 onat most 23 vertices contains a cycle of length 4 or 8, so any counterexampleto the conjecture has at least 24 vertices (previous published bound: 16);each level n = 4..23 carries a DRAT proof checked with drat-trim. (2) Thelemma underlying the 450-vertex graph of G. Exoo (arXiv:1403.5636) is false:the Tutte-Coxeter graph contains 8-cycles alternating between outer andchord edges, and the graph as specified contains 32-cycles; a correctedorientation restores the bound f(5) <= 450, with a machine-verified graph.(3) f(k) <= 15 * n_3(2^(k-2)+1) for all k >= 4, in particular f(6) <= 32640.Includes the paper, a self-contained supplement on the correction, and allgraphs (graph6/adjacency), scripts and certificates.
Source: arXiv (Cornell University)
Author: Garcia, Daniel
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223731086