Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3 (2508.19302v1)

Abstract: 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 Erd\H{o}s--Gy\'arf\'as Conjecture.

• The paper establishes that every graph with diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8, offering a constructive proof.
• It leverages a contradiction approach using local degree constraints and global diameter properties to confirm the Erdős–Gyárfás Conjecture for this class.
• The study highlights potential algorithmic applications and research avenues for detecting power-of-2 cycles in graphs under strict structural conditions.

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

Introduction

This paper establishes that any graph with diameter 2 and minimum degree at least 3 necessarily contains a cycle of length 4 or 8. The result is directly relevant to the Erdős–Gyárfás Conjecture, which asserts that every graph with minimum degree at least 3 contains a cycle whose length is a power of 2. The proof is constructive and leverages the interplay between local degree constraints and global diameter properties to guarantee the existence of such cycles.

Main Theorem and Proof Structure

The central theorem states: Let $G$ be a graph with diameter 2 and minimum degree at least 3. Then $G$ contains a cycle of length 4 or 8.

The proof proceeds by contradiction, assuming the absence of a 4-cycle and demonstrating that an 8-cycle must then exist. The argument is partitioned into two exhaustive cases based on the neighborhoods of adjacent vertices:

Case 1: Shared Neighbor ($a = c$)

If two adjacent vertices $v_1$ and $v_2$ share a neighbor $a$, the degree condition ensures the existence of additional neighbors $b$ and $d$. The absence of a 4-cycle forces $b$ and $d$ to be nonadjacent. The diameter 2 property then guarantees a vertex $v_6$ adjacent to both $b$ and $d$. If $v_6$ is not among the original vertices, a 6-cycle is formed. The diameter constraint further forces the existence of two disjoint 2-paths connecting opposite pairs, which either create a 4-cycle (contradicting the assumption) or extend the configuration to an 8-cycle.

Case 2: Distinct Neighbors ($a \neq c$)

If $a$ and $c$ are distinct and nonadjacent, the diameter 2 property ensures the existence of a vertex $x$ adjacent to both $a$ and $c$, and a vertex $y$ adjacent to both $b$ and $d$. If $x = y$, a 4-cycle is present. Otherwise, the construction yields eight distinct vertices forming an 8-cycle. The proof carefully excludes degenerate cases where $x$ or $y$ coincide with previously considered vertices, each time showing that such coincidences would force a 4-cycle.

Implications and Connections

Verification of the Erdős–Gyárfás Conjecture for Diameter 2 Graphs

The result provides a complete verification of the Erdős–Gyárfás Conjecture for the subclass of graphs with diameter 2. Specifically, it guarantees the existence of a cycle of length 4 or 8, both of which are powers of 2. This is a nontrivial strengthening, as the conjecture remains open for general graphs with minimum degree at least 3.

Structural Insights

The proof demonstrates that the combination of local degree constraints and global diameter properties imposes strong restrictions on the cycle structure of graphs. The diameter 2 condition ensures that any pair of nonadjacent vertices is connected by a path of length at most 2, which, when combined with the minimum degree, forces the existence of short cycles.

Limitations and Extensions

The result is sharp with respect to both diameter and minimum degree. For diameter greater than 2, the argument does not generalize, and for minimum degree less than 3, counterexamples exist. The constructive nature of the proof suggests algorithmic approaches for detecting such cycles in graphs with the specified properties.

Future Directions

The paper's result motivates several avenues for further research:

• Extension to Higher Diameters: Investigating whether similar cycle existence results hold for graphs with diameter greater than 2 and minimum degree at least 3.
• Algorithmic Applications: Developing efficient algorithms for finding 4- or 8-cycles in diameter 2 graphs, leveraging the constructive proof.
• Generalization to Other Powers of 2: Exploring whether analogous results can be established for cycles of length 16, 32, etc., in broader graph classes.
• Random Graph Models: Analyzing the prevalence of such cycles in random graphs with diameter 2 and prescribed minimum degree.

Conclusion

The paper rigorously proves that every graph with diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8, thereby verifying the Erdős–Gyárfás Conjecture for this class. The result is achieved through a careful case analysis that exploits the interplay between local and global graph properties. This contributes a significant step toward understanding the cycle structure of graphs under degree and diameter constraints and opens new directions for both theoretical and algorithmic research in extremal graph theory.