Paths of length five with equal-degree endpoints

Abstract: Addressing a question posed by Erdős and Hajnal, Chen and Ma proved that, for all $n \ge 600$, the complete bipartite graph $K_{n,n+1}$ is the unique graph on $2n+1$ vertices with at least $n<sup>2+n$ edges that contains no two vertices of equal degree joined by a path of length three. In this paper, we extend this result and show that, for all $n \ge 11$, $K_{n,n+1}$ is the unique $(2n+1)$-vertex graph with at least $n<sup>2+n$ edges that avoids two equal-degree vertices joined by a path of length five. This confirms the very next case of a general conjecture of Chen and Ma on paths of odd length with equal-degree endpoints.

• The paper establishes that for every n ≥ 11, the only 2n+1 vertex graph with at least n²+n edges avoiding equal-degree endpoints on a P5 is Kₙ,n+1.
• It utilizes degree partitioning, technical lemmas, and contradiction-based arguments to rigorously exclude non-bipartite extremal configurations.
• The result extends the Chen-Ma conjecture and provides a robust combinatorial toolkit for analyzing forbidden subgraph structures in dense graphs.

Extension of the Erdős–Hajnal Path Problem: Paths of Length Five with Equal-Degree Endpoints

Introduction and Background

This paper addresses the structural extremal problem in graph theory centered on the existence of same-degree vertices joined by a prescribed-length path in a dense graph. The problem traces its origin to a question of Erdős and Hajnal, who asked whether for every $(2n+1)$-vertex graph with $n^2 + n + 1$ edges, there exist two vertices of equal degree joined by a path of length three. The extremal configuration—complete bipartite graphs $K_{n, n+1}$—demonstrates the tightness of the edge bound, since such graphs avoid the property in question.

The previously settled case ($\ell = 3$) was resolved for large $n$ by Chen and Ma, and for all $n \geq 2$ by Liu and Zeng. Chen and Ma proposed the generalization that for all \emph{odd} $\ell \geq 3$ and sufficiently large $n$, the extremal graphs for the forbidden configuration (no two same-degree vertices joined by a path of length $\ell$) remain the complete bipartite case, with extremal edge count $n^2 + n$.

This work rigorously confirms the next nontrivial case $n^2 + n + 1$0, establishing both uniqueness and sharpness of the extremal configuration.

Main Result

The principal theorem establishes that for every $n^2 + n + 1$1, the only $n^2 + n + 1$2-vertex graph with at least $n^2 + n + 1$3 edges and no pair of equal-degree vertices joined by a path of length five is the complete bipartite graph $n^2 + n + 1$4. Formally, the paper resolves the value of $n^2 + n + 1$5 for these parameters, verifying the Chen-Ma conjecture for $n^2 + n + 1$6.

Methods and Technical Lemmas

The proof combines robust combinatorial enumeration with technical, degree-based partition arguments. The key methodological innovation is a reduction to degree classes coupled with extremal degree analysis, particularly focusing on duplicity and adjacency patterns among high-degree vertices. Major components include:

• A careful decomposition of vertices into sets according to neighborhoods and degree equality.
• Utilization of a technical lemma (Observation 1) giving a precise formula for the size of common neighborhoods in terms of degrees and global structure.
• Path-counting and sum-of-degrees arguments bounded by convexity and the extremal nature of bipartite graphs.

A detailed contradiction-based discharging argument is used for all non-bipartite candidate extremal structures. Using combinatorial casework and careful pairing of vertices, the methods exclude all possible structures except $n^2 + n + 1$7, leveraging the tight degree constraints in extremely dense graphs.

The proof extends to the even-vertex case in the appendix, establishing that for $n^2 + n + 1$8, the unique $n^2 + n + 1$9-vertex graph with at least $K_{n, n+1}$0 edges that avoids a path of length five between equal-degree vertices is $K_{n, n+1}$1.

Strong Quantitative and Structural Claims

• Uniqueness: For all $K_{n, n+1}$2, among all graphs with $K_{n, n+1}$3 vertices and at least $K_{n, n+1}$4 edges, only $K_{n, n+1}$5 achieves avoidance of equal-degree endpoints on a $K_{n, n+1}$6.
• Sharpness: The edge bound $K_{n, n+1}$7 is optimal; any addition creates the desired path configuration.
• No alternative extremal structure: For odd $K_{n, n+1}$8, the extremal graphs coincide with the case for $K_{n, n+1}$9.

Theoretical and Practical Implications

This result lends substantial evidence to the general conjecture for odd-length path avoidance in the context of extremal same-degree subgraph configurations. The specific identification of $\ell = 3$0 as the unique extremal structure for $\ell = 3$1 extends the established pattern for shorter odd paths. This shapes a strong inductive basis for future work on longer odd paths, suggesting that the combinatorics of extremal same-degree path presence are governed by the interplay between maximum edge counts and parity constraints on vertex classes.

On the practical side, these findings inform the design and analysis of networks where uniformity in local structure (degrees) combined with path-avoidance phenomena are critical—particularly in the construction of robust communication topologies or combinatorial design.

The technical framework, particularly the function $\ell = 3$2 and the suite of decomposition lemmas and contradiction arguments, provides a template for further attacks on forbidden subgraph problems involving degree constraints.

Future Directions

According to the concluding remarks, corresponding results for even path lengths are more intricate and do not yield the same extremal structures. Preliminary work points toward subquadratic extremal edge counts with additional structural richness, such as half-graphs for $\ell = 3$3. It is expected that the function $\ell = 3$4 exhibits markedly different behavior in the $\ell = 3$5 even and $\ell = 3$6 odd cases, opening avenues for more refined classification and extremal construction.

There is a direct invitation for future work to conclusively determine $\ell = 3$7 for even $\ell = 3$8 and large $\ell = 3$9, as well as quantifying the precise threshold for all $n$0.

Conclusion

The paper offers a definitive resolution for avoidance of paths of length five with equal-degree endpoints in dense graphs, cementing $n$1 as the unique extremal configuration for $n$2. The approach extends previous results, solidifies the odd-path Chen-Ma conjecture for the next nontrivial case, and provides a combinatorial toolkit that is likely to remain foundational for similar extremal and forbidden structure problems in discrete mathematics and theoretical computer science.