Paths of even length 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 prove that for every fixed integer (\ell\ge 2) and sufficiently large (n), the unique (2n)-vertex graph with at least ((n^2+n)/2) edges that contains no two vertices of equal degree joined by a path of length (2\ell) is the half graph (H_n). This resolves the problem posed by Chen and Ma, as well as a related question of Attwa, Azócar Carvajal, Boyadzhiyska, Pierron, and Taraz concerning paths of even length with equal-degree endpoints.

• The paper establishes that for every even length 2ℓ (with ℓ≥2) and sufficiently large n, the unique 2n-vertex graph with (n²+n)/2 edges avoiding equal-degree endpoints is the half graph Hₙ.
• It employs refined degree partitioning and combinatorial techniques to demonstrate that forbidding even-length paths enforces a strict bipartite structure.
• The work resolves conjectures by Chen and Ma and Attwa et al., setting tight Turán-type bounds with significant implications for extremal graph constructions.

Extremal Graphs Excluding Even-Length Paths with Equal-Degree Endpoints

Introduction and Problem Context

The paper "Paths of even length with equal-degree endpoints" (2607.04368) addresses a family of sharp extremal problems in graph theory originally posed by Erdős and Hajnal. The principal question is: given an $n$-vertex graph with sufficiently many edges, must there exist two vertices of equal degree connected by a path of specified even length? Prior work had resolved the case for short paths, particularly for the case of paths of length three (4-paths) in graphs on $2n+1$ vertices and odd-length generalizations, identifying complete bipartite extremal graphs $K_{n,n+1}$ as unique solutions in these settings. This paper advances the theory to cover paths of any even length and refines the structure of extremal examples in the even-order, even-length regime.

Main Contributions

The central result characterizes, for every fixed integer $\ell\ge 2$ and all sufficiently large $n$, the unique $2n$-vertex graph with at least $(n^2 + n)/2$ edges that contains no two vertices of equal degree joined by a path of length $2\ell$. This extremal graph is shown to be the half graph $H_n$, a bipartite graph introduced by Erdős and Hajnal, where the adjacency structure is determined by an ordering on partite sets; specifically, vertices $u_i$ and $2n+1$0 (for $2n+1$1) are adjacent if and only if $2n+1$2.

The paper not only settles a conjecture of Chen and Ma regarding the even-length case but also resolves a question posed by Attwa et al. on the asymptotics and extremal structure for graphs avoiding even-length paths between equal-degree endpoints.

Additionally, the authors delineate a tight lower bound for the Turán-type extremal parameter $2n+1$3, establishing that $2n+1$4 for all $2n+1$5 and large $2n+1$6, and that among $2n+1$7-vertex graphs meeting this bound, only $2n+1$8 is extremal.

Analytical and Structural Methods

The proof strategy is grounded in extremal combinatorics and structural graph theory, with extensive use of degree partitioning, path and cycle extremal results (notably Erdős–Gallai and bipartite $2n+1$9-free bounds), and a refined analysis of the induced substructures in candidate extremal graphs.

Key steps include:

• Degree Range Decomposition: Vertices are categorized based on high degree, leveraging sharp bounds on the number and structure of such vertices. Lemmas establish that only $K_{n,n+1}$0 vertices can significantly exceed degree $K_{n,n+1}$1 without introducing the forbidden configuration.
• Multiplicity Control: Forbidding $K_{n,n+1}$2-paths with equal-degree endpoints imposes strong restrictions on degree multiplicities within neighborhoods and globally. For large enough degree, carefully crafted combinatorial arguments demonstrate that degrees repeat at most a small number of times, or else produce forbidden paths.
• Structure of the Half Graph: The authors show that unless the graph closely resembles $K_{n,n+1}$3 in both degree sequence and adjacency structure, it either falls short on edge count or admits a forbidden path. This involves an iterative process of examining how large-degree vertices force specific bipartite relationships and the emergent monotonicity in the adjacency ordering.
• Uniqueness: The structure of the half graph—with strictly ordered degree sequences in both bipartite parts and no internal edges—precludes repeats of degrees along even-length paths, making $K_{n,n+1}$4 uniquely extremal.

Numerical Strength of Results and Sharpness

The extremal threshold $K_{n,n+1}$5 is shown to be tight, and the uniqueness of $K_{n,n+1}$6 in this setting is a strong claim: no other $K_{n,n+1}$7-vertex graphs meet the edge bound while maintaining the absence of even-length paths between equal-degree endpoints for large $K_{n,n+1}$8 and arbitrary $K_{n,n+1}$9.

For the base case $\ell\ge 2$0, the result matches prior work but also clarifies non-uniqueness; in contrast, for all $\ell\ge 2$1 the uniqueness phenomenon sets in. This distinction, proven via detailed combinatorial analysis, marks an important conceptual boundary in the class of extremal graph problems with forbidden degree patterns along paths.

Theoretical and Practical Implications

The results provide a definitive answer to the extremal enumeration problem in this context, identifying the precise extremal configuration and closing several open problems in the literature:

• Extremal Graph Characterization: The uniqueness of $\ell\ge 2$2 among extremal graphs gives a new characterization of the half graph in terms of local degree constraints and path structures. This adds to the known roles of the half graph in Ramsey theory, model theory, and structural graph theory.
• Further Applications: The extremal structure and techniques are likely relevant in analyzing other local-global constraints in graphs (such as colorings or induced subgraphs) and may inspire future work in forbidding other types of monochromatic or repeated patterns along paths.
• Algorithmic Considerations: Knowledge of unique extremal structures in dense settings has relevance for generating graphs with specific local/global properties (e.g., synthetic data for network analysis), and may bear on algorithms for detecting equal-degree endpoints connected by long paths.
• Generalizations: The underlying methods—degree sequence analysis, partition refinements, and extremal inequalities—are applicable to a broader class of forbidden configuration problems, potentially including directed graphs or hypergraphs, as well as paths constrained by additional local structure.

Future Directions

Open questions remain in two main directions:

• Low $\ell\ge 2$3 Behavior and Small Exceptions: For small $\ell\ge 2$4 and particular (small) $\ell\ge 2$5, the threshold and uniqueness may fail, or additional extremal graphs may arise. Complete classification in small cases and potential exceptional configurations remains an algorithmic challenge.
• Odd-Length and Generalized Forbidden Patterns: The landscape for odd-length paths and related configurations is now more fully developed, but this systematic approach suggests avenues for studying further generalizations, including other equality constraints on vertex invariants along forbidden walks or cycles.

Conclusion

This paper achieves a full structural and extremal description of $\ell\ge 2$6-vertex graphs with maximal edge count excluding even-length paths between equal-degree endpoints, identifying the half graph $\ell\ge 2$7 as unique and optimal. Techniques from extremal and structural combinatorics are pushed to their limits to realize this classification, resolving longstanding conjectures and strengthening the connection between degree pattern avoidance and bipartite extremal configurations. The methods and results have significant implications for the understanding of degree constraints in dense graphs and the architecture of extremal examples in graph theory.