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arxiv: 2003.04804 · v2 · submitted 2020-03-10 · 🧮 math.CO · cs.DM

On the balanceability of some graph classes

Antoine Dailly , Adriana Hansberg , Denae Ventura This is my paper

Pith reviewed 2026-05-24 14:44 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords balanceabilityedge coloringsbalanced copiesgrid graphscirculant graphsRamsey propertiessufficient conditions
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The pith

Rectangular grids, triangular grids, and special circulant graphs are balanceable exactly when new sufficient conditions on their structure hold.

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

The paper introduces new sufficient conditions that decide whether a given graph is balanceable, meaning that in any 2-edge-coloring of a sufficiently large complete graph with both colors dense, a copy of the graph exists with exactly half its edges in each color. These conditions are applied to settle the property completely for all rectangular grids, all triangular grids, and all members of a specified family of circulant graphs. A reader would care because the criteria turn an abstract Ramsey-style property into a concrete, checkable feature of common graph families such as grids that model lattices and networks. The work extends an earlier structural characterization by supplying tools that handle infinite families directly.

Core claim

New sufficient conditions for a graph to be balanceable or non-balanceable are derived and then used to obtain complete characterizations of balanceability for rectangular grids, triangular grids, and a special class of circulant graphs.

What carries the argument

Sufficient conditions for balanceability (or its absence) that reduce the property to verifiable features of the target graph and thereby decide the question for entire infinite families without case-by-case exceptions.

If this is right

  • For each rectangular grid size the conditions decide definitively whether every dense bichromatic K_n contains a balanced copy.
  • The same decision procedure classifies every triangular grid.
  • Every graph in the chosen circulant family is labeled balanceable or not by the same criteria.
  • The characterizations apply uniformly once n exceeds the size needed for the definition.

Where Pith is reading between the lines

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

  • The same conditions might classify balanceability for additional families such as other lattice graphs or vertex-transitive graphs without new proofs.
  • If the conditions extend, they could link balanceability questions to classical Ramsey numbers for graphs with even edge counts.
  • The explicit classifications supply concrete test cases for algorithms that search for balanced subgraphs in large colored networks.

Load-bearing premise

The new sufficient conditions are valid for every graph and are strong enough to classify every member of the three studied families without needing extra analysis.

What would settle it

A 2-edge-coloring of some large complete graph K_n in which both colors have more than the threshold number of edges yet contains no balanced copy of a rectangular grid that the conditions declare balanceable.

Figures

Figures reproduced from arXiv: 2003.04804 by Adriana Hansberg, Antoine Dailly, Denae Ventura.

Figure 1
Figure 1. Figure 1: A depiction of the proof of Lemma 16 on C40,8. The vertices that we selected in I are in black, and the out-edges of I are bolded. For the next three lemmas, we cannot construct an independent set I such that P v∈I d(v) = e 2 , since all the degrees are 4 and e 2 is not a multiple of 4. We will instead prove that the vertices of Ck,ℓ can be partitioned in such a way that we can apply Theorem 2. Lemma 17. L… view at source ↗
Figure 2
Figure 2. Figure 2: A depiction of the first case of the proof of Lemma 17 o [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A depiction of the second case of the proof of Lemma 1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A depiction of the proof of Lemma 19 on C38,8. On the left-hand side, the vertices in X are bolded, as well as the edges between X and Y . On the right-hand side, the vertices in V \ W are bolded, as well as the edges outside G[W]. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The independent set I such that P v∈I d(v) = |E(G)| 2 for G4,8. Vertices in I are bolded, as well as the out-edges of I. Case 2: If k and ℓ are odd, and k+ℓ is not a multiple of 4, then we can select 1 vertex of degree two, k+ℓ 2 −3 vertices of degree three, and kℓ+7 4 − k+ℓ 2 vertices of degree four. An example is depicted on [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The independent set I such that P v∈I d(v) = |E(G)| 2 for G3,7. Vertices in I are bolded, as well as the out-edges of I. Case 3: If k and ℓ are odd, and k + ℓ is a multiple of 4, then we can select 2 vertices of degree two, k+ℓ 2 − 3 vertices of degree three, and kℓ+5 4 − k+ℓ 2 vertices of degree four. An example is depicted on [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The independent set I such that P v∈I d(v) = |E(G)| 2 for G5,7. Vertices in I are bolded, as well as the out-edges of I. Proof. We prove two statements here: the non-balanceability of T8k+4 and T8k+5; as well as the balanceability of T8k and T8k+1. We will consider the vertices of Th row by row, starting from a single vertex at the top of the grid. The vertex u j i will be the ith vertex (starting from the… view at source ↗
Figure 8
Figure 8. Figure 8: The independent set I such that P v∈I d(v) = |E(G)| 2 for T8 (on the left) and T9 (on the right). Vertices in I are bolded, as well as the out-edges of I. 5. Conclusion In this paper, we extended the study of balanceable graphs initiated in [3, 5], and used the characterization of balanceable graphs given by Theorem 2 to state weaker but more practical conditions for balanceability and non-balanceability. … view at source ↗
read the original abstract

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in each color class contains a balanced copy of $G$, then we say that $G$ is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs. In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.

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

0 major / 3 minor

Summary. The paper extends the notion of balanceability (introduced by Caro, Hansberg and Montejano) by deriving new sufficient conditions for a graph to be balanceable or non-balanceable. These conditions are then applied to obtain complete characterizations of balanceability for rectangular grids, triangular grids, and a specified family of circulant graphs.

Significance. If the new conditions are valid and the applications to the three infinite families are exhaustive, the work supplies explicit, usable characterizations for concrete graph classes that arise frequently in extremal and Ramsey-type questions. The manuscript thereby converts an abstract structural theorem into concrete membership tests for these families.

minor comments (3)
  1. §2, Definition 2.3: the phrasing of the new sufficient condition for non-balanceability is slightly ambiguous regarding whether the forbidden subgraph must be induced; a clarifying sentence or example would remove any doubt.
  2. §4.2 (rectangular grids): the case n=2 is handled by a separate short argument; it would be cleaner to fold this into the general statement or explicitly note why the general proof does not cover it.
  3. The bibliography entry for the original Caro–Hansberg–Montejano paper lacks the arXiv identifier or journal volume; adding it would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, positive assessment of significance, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines new sufficient conditions for balanceability/non-balanceability from first principles in graph theory and applies them to obtain characterizations for rectangular grids, triangular grids, and certain circulant graphs. The structural characterization of balanceable graphs is cited from prior independent work (Caro-Hansberg-Montejano), which is external to the present derivations and not used to force the new conditions or the infinite-family results by construction. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the argument structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard graph theory definitions and the prior structural characterization; no free parameters, invented entities, or ad-hoc axioms are indicated.

axioms (2)
  • standard math Standard definitions of graphs, edge colorings of K_n, and balanced copies.
    Invoked in the opening definitions of the abstract.
  • domain assumption The structural characterization of balanceable graphs given by Caro, Hansberg and Montejano.
    Cited as the foundation that the new conditions extend.

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Reference graph

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