Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3

In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erdős-Gyárfás Conjecture by confirming it for the class of diameter-2 graphs.

💡 Research Summary

The paper addresses a special case of the Erdős‑Gyárfás conjecture, which asserts that every graph with minimum degree at least three contains a cycle whose length is a power of two. The author proves that the conjecture holds for all finite, simple, undirected graphs whose diameter is two and whose minimum degree δ(G) is at least three.

The main theorem (Theorem 1.1) states: “If G is a graph with diam(G)=2 and δ(G)≥3, then G contains a cycle of length 4 or 8.” The proof proceeds by contradiction. Assume G has no 4‑cycle. Choose an arbitrary edge v₁v₂. Because each endpoint has at least two additional neighbours, we denote the neighbours of v₁ (apart from v₂) by v₃ and v₄, and the neighbours of v₂ (apart from v₁) by v₅ and v₆. Two exhaustive cases are considered.

Case 1: v₃ = v₅.
Here v₄ and v₆ are non‑adjacent; by the diameter‑2 condition they share a common neighbour v₇. If v₇ coincides with any of the already known vertices, a 4‑cycle appears, contradicting the assumption, so v₇ is a new vertex. Repeating the same argument, v₃ and v₇ (also non‑adjacent) have a common neighbour v₈, which must be new; similarly v₈ has a neighbour v₉ distinct from the previously listed vertices. Finally, because v₉ is non‑adjacent to both v₄ and v₂, the diameter‑2 property guarantees a vertex v₁₀ that is adjacent to v₉ and to either v₄ or v₂ but not to any of the earlier vertices. In either sub‑case the vertices v₁₀‑v₂‑v₃‑v₁‑v₄‑v₇‑v₈‑v₉‑v₁₀ or v₁₀‑v₄‑v₁‑v₂‑v₆‑v₇‑v₈‑v₉‑v₁₀ form an 8‑cycle, establishing the theorem for this case.

Case 2: v₃ ≠ v₅.
First it is shown that no edge can join the sets {v₃,v₄} and {v₅,v₆}, because such an edge together with v₁v₂ would create a 4‑cycle. For each pair (a,b) with a∈{v₃,v₄} and b∈{v₅,v₆}, the diameter‑2 condition forces a common neighbour. By a counting argument at least one pair (a,b) has a common neighbour v₇ that is not v₁ or v₂; otherwise a 4‑cycle would again appear. Assume without loss of generality that a=v₃ and b=v₅, so v₇∈N(v₃)∩N(v₅) and v₇ is new. It is then proved that v₇ is not adjacent to v₄; otherwise a 4‑cycle would arise. The vertices v₄ and v₆ are also non‑adjacent, so they share a new common neighbour v₈. Several sub‑cases are examined depending on whether v₈ coincides with v₁, v₂, or is entirely new. In each sub‑case the existence of further common neighbours (by the diameter‑2 property) yields an explicit 8‑cycle, for example: v₇‑v₃‑v₁‑v₄‑v₈‑v₆‑v₂‑v₅‑v₇. Thus, if a 4‑cycle is absent, an 8‑cycle must exist.

The proof is constructive: at each step the diameter‑2 condition guarantees a common neighbour, while the minimum‑degree condition ensures that each vertex has enough unused neighbours to continue the construction without creating a forbidden 4‑cycle. Consequently, the theorem holds for all graphs satisfying the stated hypotheses.

The paper concludes with acknowledgments and a bibliography that includes the original Erdős‑Gyárfás problem, several partial results for planar, cubic, claw‑free, and P₈‑free graphs, and computational studies related to the conjecture.

Significance.
By confirming the Erdős‑Gyárfás conjecture for the whole class of diameter‑2 graphs, the work identifies a natural global constraint (diameter) that, together with a modest local constraint (δ≥3), forces the existence of very short power‑of‑two cycles. This result narrows the search for potential counterexamples to graphs of larger diameter and suggests that similar techniques—repeated exploitation of the “every non‑adjacent pair has a common neighbour” property—might be fruitful for diameter‑3 or higher, perhaps with stronger degree conditions. Moreover, the constructive nature of the proof hints at an algorithmic procedure that can locate a 4‑ or 8‑cycle in linear time, which could be of independent interest for graph‑search applications.