Recognition: no theorem link
The density of graphs with no ell-path connecting equal-degree vertices: a short proof
Pith reviewed 2026-05-12 02:16 UTC · model grok-4.3
The pith
Any graph with more than half the possible edges, for large enough size, must have two equal-degree vertices joined by a path of any fixed length.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Every sufficiently large n-vertex graph whose edge count exceeds (1/2 + o(1)) binom(n,2) contains two vertices of equal degree that are joined by a path of length ℓ.
What carries the argument
The asymptotic extremal density for the family of graphs that avoid ℓ-paths between any pair of equal-degree vertices.
If this is right
- The result determines the precise asymptotic density threshold for this forbidden configuration when the path length is odd.
- For even path lengths the same upper bound on the density still applies, but the constructions showing tightness may not reach 1/2, leaving the exact constant open.
- The proof technique directly yields that the property is forced once the edge count crosses the stated threshold, without needing additional regularity assumptions.
- The bound immediately implies that random graphs with edge probability p > 1/2 + ε contain many such equal-degree ℓ-paths with high probability.
Where Pith is reading between the lines
- The same density threshold may apply to other degree-like invariants, such as two vertices with equal numbers of common neighbors connected by an ℓ-path.
- It would be natural to ask for the exact (non-asymptotic) threshold function when ℓ is fixed and n grows, or when ℓ grows slowly with n.
- The result connects to broader questions about degree-constrained subgraphs in dense graphs, such as the existence of paths with prescribed degree sequences.
Load-bearing premise
The number of vertices is large enough depending on the fixed path length ℓ.
What would settle it
A sequence of n-vertex graphs whose edge density approaches exactly 1/2 from above yet contains no ℓ-path between equal-degree vertices, for odd ℓ.
read the original abstract
Addressing a question posed by Chen and Ma from an asymptotic point of view, we present a short proof for the edge density needed to guarantee that two vertices of the same degree are connected by a path of a fixed length. In particular, we show that for any sufficiently large graph, a density of at least $1/2+o(1)$ enforces the existence of two such vertices. This bound is tight for paths of odd length.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a short proof that any sufficiently large n-vertex graph with edge density at least 1/2 + o(1) contains two vertices of equal degree joined by a path of fixed length ℓ. The bound is tight for odd ℓ via standard constructions (e.g., balanced complete bipartite graphs or degree-partitioned graphs). The work addresses a question of Chen and Ma from an asymptotic viewpoint.
Significance. If the argument holds, the result supplies a clean asymptotic answer to the Chen-Ma question on the extremal density forcing an ℓ-path between equal-degree vertices. The shortness of the proof and the matching lower bound for odd ℓ are genuine strengths; the former may make the argument useful for follow-up work on related degree-path problems in extremal graph theory.
minor comments (2)
- The abstract and introduction would benefit from a single sentence clarifying whether ℓ is fixed independently of n or may grow slowly with n; the current phrasing leaves this slightly ambiguous even though the proof presumably treats ℓ as constant.
- A brief comparison (one paragraph) with the original Chen-Ma bounds or with related results on degree-equality in paths would help readers situate the improvement.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, including the recognition of its shortness and the matching lower bound for odd ℓ. We appreciate the recommendation to accept.
Circularity Check
No significant circularity
full rationale
The paper offers a short combinatorial proof of an asymptotic extremal statement: any sufficiently large n-vertex graph with edge density 1/2 + o(1) contains an ℓ-path whose endpoints have equal degree, for fixed ℓ. The argument relies on standard counting and density-increment techniques that do not invoke fitted parameters, self-referential definitions, or load-bearing self-citations. The question originates from external authors Chen and Ma, and tightness for odd ℓ is established via explicit bipartite or degree-partition constructions that are independent of the proof. No equation or step reduces by construction to its own inputs, so the derivation chain is self-contained.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard facts of graph theory such as the handshaking lemma and basic path-counting arguments hold.
Reference graph
Works this paper leans on
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[4]
Paths of length five with equal-degree endpoints
Paths of length five with equal-degree endpoints , author=. arXiv:2604.11664 , year=
work page internal anchor Pith review Pith/arXiv arXiv
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[6]
Graph theory, combinatorics, and applications, Vol
Problems and results in combinatorial analysis and combinatorial number theory , author=. Graph theory, combinatorics, and applications, Vol. 1 (
discussion (0)
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