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arxiv: 2605.27111 · v1 · pith:APW3F3O5 · submitted 2026-05-26 · math.CO

The list r-hued coloring of trees and unicyclic graphs

Yu Miao , Fengxia Liu This is my paper

Reviewed by Pith2026-06-29 17:00 UTCgrok-4.3pith:APW3F3O5open to challenge →

classification math.CO MSC 05C15
keywords list r-hued coloringtreesunicyclic graphschromatic numberlist coloringgraph coloring
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The pith

Trees have list r-hued chromatic number exactly min{r, Δ(G)} + 1.

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

The paper proves that trees always satisfy χ_{L,r}(G) = min{r, Δ(G)} + 1. For unicyclic graphs that are not cycles, the same equality holds when the cycle length is not five and r is at least three. In the remaining cases the value lies between min{r, Δ(G)} + 1 and min{r, Δ(G)} + 2. This extends the known non-list r-hued bound for trees to the list setting by direct case analysis on graph structure.

Core claim

If G is a tree, then χ_{L,r}(G) = min{r, Δ(G)} + 1. Let G be a unicyclic graph which is not isomorphic to the cycle C_n. If n ≠ 5 and r ≥ 3, then χ_{L,r}(G) = min{r, Δ(G)} + 1; otherwise, min{r, Δ(G)} + 1 ≤ χ_{L,r}(G) ≤ min{r, Δ(G)} + 2.

What carries the argument

The list r-hued chromatic number χ_{L,r}(G), the smallest k such that every assignment of k-lists to the vertices admits an (L,r)-coloring.

If this is right

  • The list version of the bound matches the ordinary r-hued bound exactly for every tree.
  • The same exact match holds for unicyclic graphs whose cycle length is not 5 when r ≥ 3.
  • For the cycle C_5 or when r < 3 the list number is at most one larger than the ordinary number.
  • Cycles themselves already satisfy equality between list and ordinary versions.

Where Pith is reading between the lines

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

  • The list constraint adds no extra cost for these graphs under the given structural conditions.
  • The same inductive or case-based approach may extend to graphs with few cycles or bounded treewidth.
  • Explicit list assignments that saturate the bound could be constructed to test tightness on small examples.

Load-bearing premise

The structure of trees and unicyclic graphs permits the non-list r-hued bounds to extend to arbitrary lists by case analysis without extra color demands.

What would settle it

A concrete tree or qualifying unicyclic graph together with a list assignment that forces any (L,r)-coloring to use more than min{r, Δ(G)} + 1 colors.

Figures

Figures reproduced from arXiv: 2605.27111 by Fengxia Liu, Yu Miao.

Figure 1
Figure 1. Figure 1: The graph Tvi and a1 ̸= a2, then Tvi has an (L, r)-coloring c satisfying c(vi) = a1 and c(vi−1) = c(vi+1) = a2. Proof. We prove the lemma by induction on |V (Tvi )|. Since Tvi ∈ T , by the definition of T , |V (Tvi )| ≥ 4. If |V (Tvi )| = 4, then Tvi = K1,3 and k = min{r, 3} + 1. Define NK1,3 (vi) = {v0, vi−1, vi+1}. Proof of (i). Suppose that L is an arbitrary k-list of K1,3. We consider the following cas… view at source ↗
read the original abstract

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $\chi_{r}(G)=\min\{r,\Delta(G)\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $\chi_{L,r}(C_{n})=\chi_{r}(C_{n})$. The main results are the following: $(1)$ If $G$ is a tree, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; $(2)$ Let $G$ be a unicyclic graph which is not isomorphic to the cycle $C_{n}$. If $n\neq 5$ and $r\geq3$, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; otherwise, $\min\{r,\Delta(G)\}+1\leq\chi_{L,r}(G)\leq\min\{r,\Delta(G)\}+2$.

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 / 2 minor

Summary. The paper proves two main results on list r-hued coloring: (1) every tree G satisfies χ_{L,r}(G) = min{r, Δ(G)}+1; (2) for a unicyclic graph G not isomorphic to C_n, if n ≠ 5 and r ≥ 3 then χ_{L,r}(G) = min{r, Δ(G)}+1, while in the remaining cases min{r, Δ(G)}+1 ≤ χ_{L,r}(G) ≤ min{r, Δ(G)}+2. The proofs extend the 2006 ordinary χ_r bounds via structural case analysis on trees and unicyclic graphs.

Significance. If correct, the results show that the list r-hued chromatic number coincides with the ordinary r-hued number for all trees and for unicyclic graphs except the n=5 exception (where the gap is at most 1). This is a clean extension of the cited Discrete Math 2006 theorem to the list setting and supplies explicit bounds for an infinite family of graphs.

major comments (2)
  1. [proof of result (2), unicyclic graphs with n=5] The central claim in result (2) for unicyclic graphs with n=5 rests on a case analysis that must rule out list-induced color conflicts at cycle vertices and branch points. The ordinary χ_r proof selects colors freely; when lists are arbitrary, a vertex may have its available colors depleted by prior neighbor choices in ways not covered by the ordinary cases. The manuscript must exhibit an explicit argument (or additional case) showing that +2 always suffices for every (min{r,Δ}+1)-list assignment on these graphs.
  2. [proof of result (1)] For trees (result (1)), the induction or structural argument must verify that every vertex v with a list of size min{r,Δ(G)}+1 can always choose a color satisfying the r-hued condition after its neighbors are colored, without the list restriction creating an extra demand beyond the ordinary bound. If the argument only reuses the 2006 selection without checking list intersections, it is incomplete.
minor comments (2)
  1. [abstract / introduction] The abstract states the results but does not define (L,r)-coloring or recall the precise r-hued neighborhood condition; a short preliminary section repeating these definitions would improve readability.
  2. [statement of main results] The exception clause “otherwise” in result (2) is slightly ambiguous; it should explicitly list the three subcases (n=5, r<3, and the cycle itself) for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying points where the transition from ordinary r-hued coloring to the list setting requires more explicit verification. We address each major comment below and will revise the manuscript to strengthen the arguments.

read point-by-point responses

  1. Referee: [proof of result (2), unicyclic graphs with n=5] The central claim in result (2) for unicyclic graphs with n=5 rests on a case analysis that must rule out list-induced color conflicts at cycle vertices and branch points. The ordinary χ_r proof selects colors freely; when lists are arbitrary, a vertex may have its available colors depleted by prior neighbor choices in ways not covered by the ordinary cases. The manuscript must exhibit an explicit argument (or additional case) showing that +2 always suffices for every (min{r,Δ}+1)-list assignment on these graphs.

    Authors: We agree that the existing case analysis for n=5 unicyclic graphs, while sufficient for the ordinary χ_r bound, does not explicitly track the intersection of lists with the colors forbidden by already-colored neighbors. In the revision we will insert a dedicated subsection that enumerates the possible list configurations at the cycle vertices and any pendant trees, verifying that at least one admissible color remains in each list after accounting for the at-most-r-1 forbidden colors per neighbor. This will confirm that the +2 bound holds for arbitrary lists. revision: yes

  2. Referee: [proof of result (1)] For trees (result (1)), the induction or structural argument must verify that every vertex v with a list of size min{r,Δ(G)}+1 can always choose a color satisfying the r-hued condition after its neighbors are colored, without the list restriction creating an extra demand beyond the ordinary bound. If the argument only reuses the 2006 selection without checking list intersections, it is incomplete.

    Authors: The referee correctly notes that the tree proof must explicitly confirm the existence of a suitable color inside the given list. The current manuscript reuses the counting argument from the 2006 paper (at most min{r,Δ}-1 colors are forbidden) but does not restate the intersection step. We will add a short lemma showing that when |L(v)| = min{r,Δ(G)}+1 and at most min{r,Δ(G)}-1 colors are excluded by the r-hued condition on the already-colored neighbors, L(v) always contains at least two admissible colors, guaranteeing a choice. This makes the list version self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: extends external 2006 result via structural case analysis

full rationale

The paper cites an independent 2006 Discrete Math paper for the ordinary χ_r bound on trees and states the known equality χ_{L,r}(C_n)=χ_r(C_n) for cycles. The list versions for trees and unicyclic graphs are established by direct case analysis on graph structure (trees, unicyclic with n≠5, exceptions for C_5). No self-citations are load-bearing, no parameters are fitted and renamed as predictions, and no definitions or ansatzes reduce the claimed equalities to their inputs by construction. The derivation chain remains self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, invented entities, or ad-hoc axioms beyond the standard definitions of graphs and list colorings taken from prior literature.

axioms (1)
  • standard math Standard definitions and properties of graphs, lists, and (L,r)-colorings as established in the cited 2006 Discrete Math paper
    All results are stated to rest on these background notions.

pith-pipeline@v0.9.1-grok · 5781 in / 1297 out tokens · 67195 ms · 2026-06-29T17:00:31.480921+00:00 · methodology

discussion (0)

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Reference graph

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