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arxiv: 1906.10031 · v1 · pith:Y5W6SCMAnew · submitted 2019-06-24 · 🧮 math.CO · cs.DM

Hierarchical Colorings of Cographs

D.I. Valdivia , M. Gei{\ss} , M. Hellmuth , M. Hernandez Rosales , P.F. Stadler This is my paper

Pith reviewed 2026-05-25 17:33 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords cographshierarchical coloringgreedy coloringchromatic numberhereditarily well-colored graphsinduced subgraphs
0
0 comments X

The pith

Cographs can be hierarchically colored using no more than their chromatic number of colors.

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

Cographs are the graphs where every induced subgraph admits a greedy coloring with exactly χ(G) colors. The paper shows that hierarchical colorings, introduced in work on reciprocal best match graphs, treat greedy colorings as one special case within a larger family. It proves that this family also uses at most χ(G) colors on every induced subgraph of a cograph. The result therefore extends the known characterization of cographs from greedy colorings alone to the hierarchical setting. A reader cares because the same graphs that behave well under one coloring rule continue to do so under the more general rule.

Core claim

Cographs are exactly the hereditarily well-colored graphs, meaning a greedy coloring of any induced subgraph uses only χ(G) colors. Hierarchical colorings generalize greedy colorings, and the same hereditary property holds: every induced subgraph of a cograph admits a hierarchical coloring that also uses at most χ(G) colors.

What carries the argument

Hierarchical coloring, the generalization of greedy coloring that admits a hierarchical structure on color assignments while still requiring at most χ(G) colors on cographs.

Load-bearing premise

The definition of hierarchical coloring makes greedy colorings a special case inside it, and the minimal-color property of cographs carries over unchanged to this wider class.

What would settle it

An explicit cograph together with one of its induced subgraphs and a hierarchical coloring of that subgraph that uses more than χ(G) colors.

Figures

Figures reproduced from arXiv: 1906.10031 by D.I. Valdivia, M. Gei{\ss}, M. Hellmuth, M. Hernandez Rosales, P.F. Stadler.

Figure 1
Figure 1. Figure 1: The induced cotree of a cograph G affects the hc-coloring property of σ, where σ(a) = σ(d) 6= σ(b) = σ(c). In (T, t), the first tree from left to right, (K1)-(K3) are satisfied making σ an hc-coloring. However the second tree (T 0 , t0 ) does not satisfy (K3) since the parent node of c ' K1 and d ' K1 corresponds to a disjoint union operation and σ(c) ∩ σ(d) = ∅. Thus σ is not an hc-coloring w.r.t. (T 0 , … view at source ↗
Figure 2
Figure 2. Figure 2: Both colorings A and B of K2∪· K1∪· K1 are hc-colorings. However, only A is a greedy coloring since the two single-vertex components have different colors in B. First consider G = G1 O G2. Since xy ∈ E(G) for all x ∈ V1 and y ∈ V2 we have σ(x) 6= σ(y), and hence σ(V1) ∩ σ(V2) = ∅. Thus, σ(V ) = σ(V1) ∪· σ(V2) and therefore, |σ(V )| = |σ(V1)| + |σ(V2)| IH. = χ(G1) + χ(G2) Equ. (1) = χ(G). We note that the c… view at source ↗
read the original abstract

Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $\chi(G)$. In recent work on reciprocal best match graphs so-called hierarchically coloring play an important role. Here we show that greedy colorings are a special case of hierarchical coloring, which also require no more than $\chi(G)$ colors.

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

2 major / 0 minor

Summary. The manuscript states that cographs are precisely the hereditarily well-colored graphs (every induced subgraph admits a greedy coloring with exactly χ(G) colors). It then claims that greedy colorings form a special case of the hierarchical colorings introduced in recent work on reciprocal best match graphs, and therefore that every hierarchical coloring of a cograph also uses at most χ(G) colors.

Significance. If the claimed embedding of greedy colorings inside the hierarchical-coloring framework is valid, the result would transfer the hereditary well-coloring property of cographs to hierarchical colorings, supplying an immediate χ(G) bound for a coloring notion that arises in the RBMG literature. The manuscript supplies no new combinatorial machinery beyond this observation.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'greedy colorings are a special case of hierarchical coloring' is asserted without reproducing the definition of hierarchical coloring (imported from the RBMG literature) or exhibiting an explicit embedding of the greedy algorithm inside that definition. Without this step the asserted bound ≤ χ(G) for hierarchical colorings rests on an unverified external assumption.
  2. [Abstract] Abstract: no proof or verification steps are supplied for the statement that hierarchical colorings of cographs require no more than χ(G) colors. The hereditary well-coloring property is recalled but not shown to transfer once the larger class of hierarchical colorings is admitted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for pointing out these issues in the presentation of our results. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'greedy colorings are a special case of hierarchical coloring' is asserted without reproducing the definition of hierarchical coloring (imported from the RBMG literature) or exhibiting an explicit embedding of the greedy algorithm inside that definition. Without this step the asserted bound ≤ χ(G) for hierarchical colorings rests on an unverified external assumption.

    Authors: The referee correctly notes that the abstract does not reproduce the definition of hierarchical colorings or provide an explicit embedding. Since the definition is imported from the referenced RBMG literature, the current manuscript assumes reader familiarity with that work. We will revise the manuscript to include the definition of hierarchical colorings and an explicit demonstration that greedy colorings constitute a special case within this framework. revision: yes

  2. Referee: [Abstract] Abstract: no proof or verification steps are supplied for the statement that hierarchical colorings of cographs require no more than χ(G) colors. The hereditary well-coloring property is recalled but not shown to transfer once the larger class of hierarchical colorings is admitted.

    Authors: We agree that the manuscript does not supply detailed proof steps for the bound on the number of colors used by hierarchical colorings of cographs. The hereditary well-coloring property is stated, but the transfer to hierarchical colorings is presented as following from the special case relation without further elaboration. In the revised version, we will include the necessary verification or proof to show that the bound holds for hierarchical colorings as well. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit demonstration of special case uses external definition without reduction to inputs

full rationale

The paper states a known fact about cographs (hereditarily well-colored, i.e., greedy colorings use χ(G) colors on induced subgraphs) and imports the definition of hierarchical coloring from prior RBMG literature. It then claims to demonstrate directly that greedy colorings are a special case within that definition, from which the χ(G) bound transfers. No equations, parameters, or premises reduce by construction to fitted inputs or self-citations; the load-bearing step is the explicit showing of the embedding, which is presented as the paper's contribution rather than an unexamined assumption. This is self-contained against the external benchmark of the imported definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on the known characterization of cographs as hereditarily well-colored graphs and on the definition of hierarchical coloring from cited recent work on reciprocal best match graphs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Cographs are exactly hereditarily well-colored graphs
    Stated directly in the abstract as the starting point for the result.

pith-pipeline@v0.9.0 · 5598 in / 1001 out tokens · 27747 ms · 2026-05-25T17:33:03.497856+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Complexity of Modification Problems for Reciprocal Best Match Graphs

    cs.CC 2019-07 unverdicted novelty 6.0

    Deletion and editing to RBMGs are NP-hard; 2-colored editing is FPT via bicluster editing reduction; modification to hierarchically colored cographs is NP-complete.

Reference graph

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