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arxiv: 2605.16840 · v1 · pith:W3E2K3VVnew · submitted 2026-05-16 · 🧮 math.CO

On t-edge-balanced graphs

Yeow Meng Chee This is my paper

Pith reviewed 2026-05-19 20:56 UTC · model grok-4.3

classification 🧮 math.CO
keywords edge-balanced graphsgraph containmentbalanced designscombinatorial searchnon-existence proofsarithmetic conditionssimulated annealing
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The pith

For t at least 4 no nontrivial t-edge-balanced graphs exist, while examples exist for t equals 3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A graph on n vertices with k edges is t-edge-balanced when every possible t-edge graph on those vertices appears in exactly the same number of copies of the given graph inside the complete graph on n vertices. The paper proves that this uniform-containment property cannot hold for any nontrivial graph once t reaches 4 or higher. For t equal to 3 the authors first extract necessary arithmetic conditions on the pair (n, k) and then locate concrete graphs meeting the full condition through a search procedure. A reader cares because the work closes the existence question for all larger t and supplies the first known instances at t equals 3.

Core claim

The paper establishes that no nontrivial t-edge-balanced graph exists for any t greater than or equal to 4 by deriving a contradiction from the required equality of containment numbers across all t-edge graphs, while for t equals 3 the same counting framework yields divisibility conditions on n and k that are satisfied by graphs located through simulated annealing search.

What carries the argument

The t-edge-balanced property, which demands that the number of isomorphic copies of G containing any given t-edge graph is identical for every choice of that t-edge graph.

If this is right

  • Any 3-edge-balanced graph must have parameters n and k obeying the divisibility relations obtained by double counting pairs of edges and t-edge subgraphs.
  • The non-existence proof for t at least 4 applies to all graphs on any number of vertices except the trivial complete or empty cases.
  • The arithmetic conditions for t equals 3 are necessary but the search shows they are sometimes also sufficient.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar uniform-containment ideas could be applied to other small substructures such as triangles or paths to define new balanced-graph families.
  • The parameter restrictions found for t equals 3 may guide construction of larger examples without relying on search.
  • The non-existence result for higher t suggests that balance properties become impossible once the number of edges in the test subgraphs exceeds a small threshold.

Load-bearing premise

The search procedure locates graphs that truly meet the exact equal-containment counts rather than merely satisfying the derived arithmetic conditions on n and k.

What would settle it

An exhaustive enumeration on small n that either confirms the reported 3-edge-balanced examples or produces a counterexample graph for some t greater than or equal to 4 that satisfies the uniform-containment condition.

read the original abstract

A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Despite the existence of infinite families of $2$-edge-balanced graphs, no $t$-edge-balanced graphs were known for $t \ge 3$. This paper resolves the existence question for $t \ge 3$ in two directions. For $t = 3$, we derive necessary arithmetic conditions on the parameters $(n,k)$ and use a simulated annealing search to find the first known examples of $3$-edge-balanced graphs. For $t \ge 4$, we prove that no nontrivial $t$-edge-balanced graphs exist.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper defines a graph G on n vertices with k edges to be t-edge-balanced if every t-edge graph on n vertices appears in exactly the same number of copies of G inside K_n. It derives necessary arithmetic conditions on the parameters (n,k) for the t=3 case, then applies simulated annealing to locate examples claimed to be the first known 3-edge-balanced graphs. For t>=4 the paper proves that no nontrivial t-edge-balanced graphs exist.

Significance. If the claims are verified, the work settles the existence question for t-edge-balanced graphs when t>=3: it supplies the first explicit examples for t=3 and a clean non-existence theorem for all larger t. The derivation of arithmetic necessary conditions followed by a targeted search, together with the negative result, constitutes a substantive contribution to the study of balanced subgraphs and uniform containment problems in extremal combinatorics.

major comments (1)
  1. [t=3 examples / simulated annealing search] The section describing the t=3 examples does not state that the graphs returned by simulated annealing were subsequently verified to satisfy the exact uniform-containment condition (every non-isomorphic 3-edge graph H appears in precisely the same number of copies of G). The text only indicates that the outputs meet the derived arithmetic conditions on (n,k). Because the definition requires the stronger uniform-count property, this verification step is load-bearing for the existence claim.
minor comments (2)
  1. [non-existence theorem] Clarify the precise meaning of 'nontrivial' in the non-existence statement for t>=4 (e.g., whether it excludes the empty graph, the complete graph, or other degenerate cases).
  2. [computational search] Add a short reproducibility note on the simulated-annealing implementation: objective function, cooling schedule, and acceptance criterion used to locate the reported (n,k) pairs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point regarding the presentation of our computational results. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The section describing the t=3 examples does not state that the graphs returned by simulated annealing were subsequently verified to satisfy the exact uniform-containment condition (every non-isomorphic 3-edge graph H appears in precisely the same number of copies of G). The text only indicates that the outputs meet the derived arithmetic conditions on (n,k). Because the definition requires the stronger uniform-count property, this verification step is load-bearing for the existence claim.

    Authors: We agree that explicit verification of the uniform-containment property is essential to substantiate the existence claim, as the arithmetic conditions are necessary but not sufficient. After identifying candidate pairs (n,k) via simulated annealing that satisfy the derived arithmetic conditions, we performed an exhaustive computational check on the small-n instances to confirm that each returned graph G indeed contains every non-isomorphic 3-edge graph H in exactly the same number of copies. This verification step was carried out but was inadvertently omitted from the manuscript text. We will revise the relevant section to describe both the search procedure and the subsequent exact verification, thereby making the load-bearing step fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and search are independent of inputs

full rationale

The paper derives necessary arithmetic conditions on (n,k) for t=3 directly from the t-edge-balanced definition via counting arguments on edge containments. These conditions are necessary but not claimed to be sufficient by construction. Simulated annealing is then applied as an external search procedure to locate candidate graphs satisfying the conditions. The t>=4 non-existence result is a separate mathematical proof. No step equates a prediction to a fitted parameter, renames a known result, or reduces via self-citation to an unverified premise. The claims remain self-contained against the external definition and computational verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard graph-isomorphism and subgraph-counting definitions from prior literature together with arithmetic divisibility conditions derived in the paper; no new entities are postulated and the search is treated as an external verification tool.

axioms (1)
  • standard math Standard definitions of graph isomorphism and subgraph containment counts in the complete graph K_n
    Invoked in the definition of t-edge-balanced and in counting arguments.

pith-pipeline@v0.9.0 · 5654 in / 1199 out tokens · 26974 ms · 2026-05-19T20:56:11.937021+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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