What question did this study set out to answer?

The research aims to establish cycle length bounds in honeycomb toroidal graphs of girth 6 according to the Erdos–Gyárfás conjecture.

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February 8, 2026Open Access

A Mechanism Dichotomy for Power-of-Two Cycles in Honeycomb Toroidal Graphs of Girth 6

Key Points

The research aims to establish cycle length bounds in honeycomb toroidal graphs of girth 6 according to the Erdos–Gyárfás conjecture.
Proved the Erdos-Gyárfás conjecture for honeycomb toroidal graphs of girth 6.
Developed a Mechanism Dichotomy framework for cubic bipartite graphs.
Classified cases using column-span analysis and cycle spectrum results.
Demonstrated that the smallest exponent k for cycles in HTGs satisfies kmin(G) ≤ 4.
Identified cases leading to the existence of 8-cycles and 16-cycles based on the structure of the graphs.

Abstract

We prove that the Erdos–Gyárfás as conjecture holds for all honeycomb toroidal graphs (HTGs) of girth 6, establishing that the smallest exponent k such that Gcontains a cycle of length 2k satisfies kmin(G) ≤4. Our main tool is a novel Mechanism Dichotomy framework for cubic bipartite graphs of girth 6: Case 1: Tight pairs of hexagons exist ⇒8-cycles via symmetric difference (Thetaresonance). Case 2a: No tight pairs, but 8-cycles exist independently. Case 2b: No tight pairs and no 8-cycles ⇒16-cycles via Alspach’s bypass construction. For HTGs, we fully classify these cases using column-span analysis and Alspach’s cycle spectrum results.

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A Mechanism Dichotomy for Power-of-Two Cycles in Honeycomb Toroidal Graphs of Girth 6

Key Points

Abstract

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