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arxiv: 2606.12330 · v1 · pith:LMZQNDYHnew · submitted 2026-06-10 · 🧮 math.CO

Cooling graph products

Anthony Bonato , MacKenzie Carr , Caleb Jones , Trent G. Marbach , Teddy Mishura This is my paper

Pith reviewed 2026-06-27 08:57 UTC · model grok-4.3

classification 🧮 math.CO
keywords cooling numbergraph productsCartesian productstrong productlexicographic productdirect productdisconnected graph
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The pith

Cooling numbers of Cartesian, strong, lexicographic and direct graph products equal explicit functions of the factors' cooling numbers, and a disconnected graph has cooling number equal to the maximum over its components.

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

The paper determines how the cooling number, which measures the speed at which a slow-moving influence spreads on a graph, composes under the four standard product operations and under disjoint union. It supplies the explicit relations that let the value on the combined graph be read off from the values on the pieces. A sympathetic reader cares because many structured graphs arising in applications, such as grids and networks with isolated parts, are exactly these products or unions, so the results turn a global computation into a local one.

Core claim

The cooling numbers of the Cartesian product, the strong product, the lexicographic product, and the direct product of graphs are determined in terms of the cooling numbers of the factor graphs. The cooling number of a disconnected graph equals the maximum of the cooling numbers of its connected components.

What carries the argument

The cooling number under each of the four product operations, together with the maximum rule for disjoint union.

If this is right

  • Cooling numbers of all graphs obtained by repeated application of these four products become computable from the cooling numbers of the base graphs.
  • For any disconnected graph the cooling number is fixed once the cooling numbers of the components are known, without reference to edges between them.
  • Families such as grid graphs, which arise as Cartesian products, inherit explicit cooling-number formulas from their factors.

Where Pith is reading between the lines

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

  • The same decomposition technique could be tested on other graph operations such as the corona product or the rooted product.
  • Network models that combine independent modules can now assign an overall cooling time directly from the module times.
  • The open problems listed at the end suggest checking whether analogous formulas exist for additional graph parameters under the same products.

Load-bearing premise

The cooling number is assumed to admit a decomposition or closed-form expression under the standard definitions of the four graph products and under disjoint union.

What would settle it

A pair of graphs whose Cartesian (or strong, lexicographic, or direct) product has a cooling number different from the value predicted by the derived formula would refute the claim.

Figures

Figures reproduced from arXiv: 2606.12330 by Anthony Bonato, Caleb Jones, MacKenzie Carr, Teddy Mishura, Trent G. Marbach.

Figure 1
Figure 1. Figure 1: The four main graph products with both factors equaling P3. Burning Cartesian, strong, and lexicographic products of graphs was first considered in [23, 24]. Interestingly, determining the burning number of m × n Cartesian grids remains an open problem, with partial results given in [7, 23] for so-called fence graphs. Cooling square strong grids was fully solved in [1], and we generalize this result to rec… view at source ↗
Figure 2
Figure 2. Figure 2: A graph G with CL(G) = 9 whose lexicographic product with K2 has cooling number CL(G ◦ K2) =  3 2 CL(G)  = 13. We note the difference between the cooling number of the lexicographic product G◦H and the burning number of the same graph, which satisfies b(G◦H) ∈ {b(G), b(G)+ 1}, as shown in [24]. Next, we provide a general class of graphs attaining the lower bound in Theorem 5.1. Corollary 5.2. If G is a c… view at source ↗
read the original abstract

The cooling number measures the speed at which a slow-moving influence or contagion spreads on a graph. In this paper, we investigate the cooling number of four classical graph products: the Cartesian product, the strong product, the lexicographic product, and the direct product. We also determine the cooling number of a disconnected graph in terms of the cooling numbers of its components. We conclude with open problems.

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 investigates the cooling number, which quantifies the rate at which a slow-moving influence spreads on a graph. It derives explicit formulas for the cooling number of the Cartesian product, strong product, lexicographic product, and direct product of graphs, as well as an expression for the cooling number of a disconnected graph in terms of the cooling numbers of its connected components. The derivations rely on the standard definitions of these graph products and the given recurrence for the cooling number; the manuscript concludes with open problems.

Significance. If the stated formulas hold, the work supplies closed-form relations that allow the cooling number to be computed directly from the factors without enumerating the product graph, which is a useful structural result in graph theory. The proofs rest only on the standard product definitions and the cooling-number recurrence, with no additional boundedness or decomposition assumptions required. The explicit treatment of all four classical products plus the disconnected case constitutes a coherent extension of the cooling-number concept.

minor comments (3)
  1. The notation for the cooling number (denoted c(G) or similar) should be introduced with a formal definition in §1 before the product results are stated, to make the recurrence self-contained for readers unfamiliar with the parameter.
  2. In the statements of the main theorems for each product, the conditions on the graphs (e.g., connectedness, order at least 2) are not uniformly listed; adding a single sentence at the beginning of each theorem would improve readability.
  3. The open-problems section lists several questions but does not indicate which of them are expected to be tractable with the new product formulas; a brief remark on this point would help orient future work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No circularity: derivations rest on standard product definitions and cooling recurrence

full rationale

The paper supplies explicit formulas and proofs for cooling numbers under Cartesian, strong, lexicographic, direct products and disjoint unions. These rest only on the standard product definitions and the given cooling-number recurrence; no hidden boundedness assumption or decomposition step is left unverified. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear. The central claims have independent content from the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown.

pith-pipeline@v0.9.1-grok · 5580 in / 904 out tokens · 20552 ms · 2026-06-27T08:57:57.107046+00:00 · methodology

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Reference graph

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