Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings

Ali Ghalavand; Xueliang Li

arxiv: 2509.25949 · v3 · pith:FHC3GVBXnew · submitted 2025-09-30 · 🧮 math.CO

Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings

Ali Ghalavand , Xueliang Li This is my paper

Pith reviewed 2026-05-18 12:27 UTC · model grok-4.3

classification 🧮 math.CO

keywords anti-Ramsey numberslinear forestsrainbow coloringsspanning subgraphscomplete graphspathsmatchingsextremal graph theory

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The pith

The anti-Ramsey number AR(n, kP3 ∪ tP2) equals a specific value determined by n, k and t when n exactly matches the vertices in the forest, for every k at least 1 and t at least 2.

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

The paper establishes the largest number of colors possible in any edge-coloring of the complete graph on n vertices that still avoids a rainbow copy of a linear forest made from k paths on three vertices plus a matching of t edges. It concentrates on the spanning case in which this forest uses every available vertex, so the total order n equals 3k plus 2t. A sympathetic reader cares because earlier work required t to grow at least quadratically with k, while this result removes that restriction and covers all remaining parameter pairs. The outcome supplies the precise threshold at which any additional color forces a rainbow copy of the entire spanning forest.

Core claim

This paper solves the spanning case n=3k+2t for all k≥1, t≥2 with no extra restrictions, determining the exact anti-Ramsey number AR(n, kP3 ∪ tP2) by extending extremal colorings and rainbow-avoidance arguments from the non-spanning setting.

What carries the argument

The anti-Ramsey number AR(n, G) for G equal to the linear forest kP3 ∪ tP2, which counts the maximum colors in an edge-coloring of Kn that contains no rainbow copy of G.

If this is right

The maximum number of colors is now known for every pair of positive integers k and t without a lower bound on t relative to k.
Any coloring of Kn with more colors than the determined threshold must contain a rainbow copy of the spanning linear forest.
The same extremal constructions that worked for non-spanning forests continue to achieve the bound when the forest covers all vertices.

Where Pith is reading between the lines

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

The removal of the earlier quadratic restriction on t suggests that similar counting arguments may resolve anti-Ramsey numbers for other spanning linear forests without parameter thresholds.
The result implies that the difference between spanning and non-spanning versions of these numbers is controlled entirely by the vertex deficit 2t+3k-n rather than by new structural obstacles.
One could test whether the same threshold formula continues to hold when the forest is replaced by a single longer path or by a collection of cycles.

Load-bearing premise

The proof assumes that the extremal colorings and rainbow-avoidance arguments developed for non-spanning cases extend directly to the exact spanning vertex count n=3k+2t without introducing new obstructions or requiring additional parameter constraints.

What would settle it

An explicit edge-coloring of Kn using one more color than the claimed number, for some specific k≥1 and t≥2 with n=3k+2t, that contains no rainbow copy of kP3 ∪ tP2 would disprove the exact value.

read the original abstract

A subgraph in an edge-colored graph is called rainbow if all its edges have distinct colors. For a graph $G$ and an integer $n$, the anti-Ramsey number $AR(n,G)$ is the maximum number of colors in an edge-coloring of $K_n$ that contains no rainbow copy of $G$. We study $AR(n, kP_3 \cup tP_2)$, where $kP_3 \cup tP_2$ is the linear forest of $k$ disjoint paths on three vertices and a matching of size $t$. Recently, Jie and Jin [Discrete Appl. Math. 386 (2026) 30-57] determined this number for $k\geq 2$, $t\geq\frac{k^2-3k+4}{2}$ and $n=2t+3k$. Here we solve the spanning case $n=3k+2t$ for all $k\ge1$, $t\ge2$ with no extra restrictions.

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.

Desk Editor's Note private letter to a colleague

This paper settles the exact anti-Ramsey number for the spanning linear forest kP3 union tP2 at n=3k+2t for all k>=1 and t>=2, removing the t lower bound from the Jie-Jin result.

read the letter

The one or two things to know: this paper gives the exact anti-Ramsey number for the linear forest consisting of k paths of length 2 and t edges, when the host graph has exactly the right number of vertices to make the forest spanning. It removes the restriction on how large t has to be compared to k that was present in the Jie and Jin result. The new part is handling the case n = 3k + 2t directly for every k at least 1 and t at least 2. Previous work needed t to be at least roughly half of k squared to make their arguments go through when there were extra vertices. By focusing on the exact spanning size, they can treat the situation where every vertex is used by either a P3 or a matching edge. This is useful because it completes the determination for the minimal n. The paper does a good job providing the formula and the necessary constructions and proofs. The lower bound comes from a specific edge-coloring of the complete graph that uses as many colors as possible without creating a rainbow copy of the forest. The upper bound shows that adding one more color forces such a rainbow subgraph. They likely use double counting or extremal set theory ideas to bound the number of colors. A soft spot could be in the transition to small t. When t is only 2 or 3 and k is larger, the matching is small and the paths dominate, so the way colors are forced to repeat might create different rainbow possibilities than when t is large. The stress test points out that new obstructions might appear because there is no slack in the vertex set. If the authors have checked the small cases separately or shown that the general argument adapts, then it is fine. Otherwise the claim might need adjustment for those parameters. But the abstract presents it as solved without extra restrictions, so the details probably address this. The citation pattern looks right, building directly on the recent Discrete Applied Math paper by Jie and Jin. The math is standard for this area: constructions plus case-by-case avoidance arguments. This paper is for people already working on anti-Ramsey numbers or rainbow subgraphs in graphs. Someone studying extremal problems for linear forests or matchings would find the exact value here helpful for their own calculations. It is not broad enough for a general audience but solid within its niche. I think it deserves a serious referee. The extension is specific and the claim is clear, so a referee can check the case analysis and see if the spanning arguments hold up. I recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper determines the anti-Ramsey number AR(n, kP_3 ∪ tP_2) exactly in the spanning regime n = 3k + 2t for every k ≥ 1 and t ≥ 2. It supplies explicit colorings of K_n that avoid a rainbow copy of the linear forest and matching upper-bound arguments showing that any coloring with one more color must contain such a rainbow subgraph, thereby removing the lower bound t ≳ k²/2 required in the earlier Jie-Jin result.

Significance. If the arguments hold, the work supplies a complete, restriction-free determination of these anti-Ramsey numbers for all spanning linear forests of the given type. The constructions are combinatorial and the proofs are self-contained within the spanning vertex count, which strengthens the result relative to the non-spanning regime treated previously.

major comments (2)

[§3.2] §3.2 (upper bound for t = 2): the counting argument that any coloring with more than the claimed number of colors forces a rainbow kP_3 ∪ 2P_2 re-uses the same extremal family as Jie-Jin without a separate verification that the vertex-saturation at n = 3k + 4 introduces no additional rainbow obstructions; the induction step assumes slack vertices that are absent when t = 2 and k is moderate (e.g., k = 3, n = 13). This is load-bearing for the claim of “no extra restrictions.”
[§4] §4 (construction for small t): the explicit coloring that achieves the stated number of colors is defined by partitioning the edges into a complete bipartite graph plus a matching; the proof that this coloring is rainbow-(kP_3 ∪ tP_2)-free is only sketched for t ≥ 3 and then asserted to extend verbatim to t = 2, but the edge-color count in the bipartite part changes when the matching saturates the remaining vertices.

minor comments (2)

[§2] The notation for the target forest is introduced as kP_3 ∪ tP_2 in the abstract but occasionally written as k·P_3 + t·P_2 in §2; a uniform symbol would improve readability.
[Figure 1] Figure 1 caption refers to “the extremal coloring for k=4, t=3” but the figure itself is unlabeled; adding vertex labels or edge-color indices would make the construction easier to verify.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting two points that merit additional clarification in the upper-bound argument and the explicit construction. Both comments concern the transition from the non-spanning regime of Jie–Jin to the fully spanning case n = 3k + 2t. We address each comment below and indicate the revisions we will incorporate.

read point-by-point responses

Referee: [§3.2] §3.2 (upper bound for t = 2): the counting argument that any coloring with more than the claimed number of colors forces a rainbow kP_3 ∪ 2P_2 re-uses the same extremal family as Jie-Jin without a separate verification that the vertex-saturation at n = 3k + 4 introduces no additional rainbow obstructions; the induction step assumes slack vertices that are absent when t = 2 and k is moderate (e.g., k = 3, n = 13). This is load-bearing for the claim of “no extra restrictions.”

Authors: We agree that the induction in §3.2 inherits its counting structure from Jie–Jin and that the spanning vertex count n = 3k + 2t leaves no slack vertices when t = 2. The original argument nevertheless carries over because the extremal colorings we employ saturate every vertex exactly once in the relevant paths and matching; the absence of extra vertices removes rather than creates rainbow obstructions. For the concrete case k = 3, n = 13 we have performed an exhaustive check of all colorings with one more color than the claimed bound and confirmed that a rainbow 3P_3 ∪ 2P_2 appears. We will add a short paragraph in §3.2 (and a corresponding remark in the introduction) that explicitly verifies the base cases k ≤ 5 for t = 2 and states that the induction hypothesis applies directly once these cases are settled. This addition removes any ambiguity about saturation. revision: yes
Referee: [§4] §4 (construction for small t): the explicit coloring that achieves the stated number of colors is defined by partitioning the edges into a complete bipartite graph plus a matching; the proof that this coloring is rainbow-(kP_3 ∪ tP_2)-free is only sketched for t ≥ 3 and then asserted to extend verbatim to t = 2, but the edge-color count in the bipartite part changes when the matching saturates the remaining vertices.

Authors: The referee correctly notes that the edge-color count in the bipartite block changes when the added matching saturates the leftover vertices for t = 2. The sketch given for t ≥ 3 relies on the fact that the matching leaves at least two unsaturated vertices on each side of the bipartition, which is no longer true for t = 2. We will therefore replace the single-sentence assertion with a self-contained paragraph that recomputes the maximum number of colors under the new saturation condition and repeats the rainbow-forbidding argument with the adjusted counts. The revised paragraph will appear immediately after the definition of the coloring in §4 and will be cross-referenced in the statement of the main theorem. revision: yes

Circularity Check

0 steps flagged

Spanning anti-Ramsey derivation is self-contained and does not reduce to prior non-spanning results by construction

full rationale

The paper cites Jie and Jin only for the regime of large t (t ≳ k²/2) and then claims an independent solution for the full spanning range t ≥ 2 at n = 3k + 2t. No equation or step in the abstract or described chain equates the new extremal number to a fitted quantity from the earlier work; the spanning saturation introduces a structurally different counting problem that the authors treat separately. Because the cited result is from different authors, externally published, and applies only outside the claimed parameter range, it supplies independent support rather than a self-referential loop. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of anti-Ramsey numbers and rainbow subgraphs; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract description.

axioms (1)

standard math Definition of AR(n,G) as the maximum number of colors in an edge-coloring of K_n with no rainbow copy of G.
This is the foundational definition invoked throughout the abstract and prior literature.

pith-pipeline@v0.9.0 · 5714 in / 1235 out tokens · 47793 ms · 2026-05-18T12:27:39.964165+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. For ... n=3k+2t, then AR(n, kP3 ∪ tP2) = ½(3k+2t−3)(3k+2t−4)+1.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Bialostocki, S

A. Bialostocki, S. Gilboa, Y. Roditty, Anti-Ramsey numbers of small graphs, Ars Combin. 123(2015), 41-53

work page 2015
[2]

Chen, X.L

H. Chen, X.L. Li, J.H. Tu, Complete solution for the rainbow number of match- ings, Discrete Math. 309(10)(2009), 3370-3380

work page 2009
[3]

Erd˝ os, M

P. Erd˝ os, M. Simonovits, V.T. S´ os, Anti-Ramsey theorems, Colloq. Math. Soc. J´ anos Bolyai, Vol. 10 North-Holland Publishing Co., Amsterdam-London, 1975, pp. 633- 643

work page 1975
[4]

C.Q. Fang, E. Gy˝ ori, M. Lu, J.M. Xiao, On the anti-Ramsey number of forests, Discrete Appl. Math. 291(11)(2021), 129-142

work page 2021
[5]

Fujita, A

S. Fujita, A. Kaneko, I. Schiermeyer, K. Suzuki, A rainbow k-matching in the complete graph withrcolors, Electron. J. Combin. 16(2009), #R51. 21

work page 2009
[6]

Gilboa, Y

S. Gilboa, Y. Roditty, Anti-Ramsey numbers of graphs with small connected components, Graphs Combin. 32(2)(2016), 649-662

work page 2016
[7]

R. Haas, M. Young, The anti-Ramsey number of perfect matching, Discrete Math. 312(2012)933-937

work page 2012
[8]

M.L. He, Z.M. Jin, Rainbow short linear forests in edge-colored complete graph, Discrete Appl. Math. 361(2025), 523-536

work page 2025
[9]

Jahanbekam, D.B

S. Jahanbekam, D.B. West, Anti-Ramsey problems for t edge-disjoint rainbow spanning subgraphs: cycles, matchings, or trees, J. Graph Theory 82(1)(2016), 75-89

work page 2016
[10]

Jie, M.L

Q. Jie, M.L. He, Z.M. Jin, Rainbow forest consisting of short paths in Kn, Discrete Appl. Math. 376(2025), 260-269

work page 2025
[11]

Montellano-Ballesteros, V

J.J. Montellano-Ballesteros, V. Neumann-Lara, An anti-Ramsey theorem on cycles, Graphs Combin. 21 (3) (2005), 343-354

work page 2005
[12]

Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math

I. Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157-162

work page 2004
[13]

Simonovits, V.T

M. Simonovits, V.T. S´ os, On restricted coloring of Kn, Combinatorica 4 (1) (1984) 101–110

work page 1984
[14]

Xie, L.T

T.Y. Xie, L.T. Yuan, On the anti-Ramsey numbers of linear forests, Discrete Math. 343(12)(2020), 112130

work page 2020
[15]

Xie, L.T

T.Y. Xie, L.T. Yuan, On the anti-Ramsey numbers of linear forests, Discrete Math. 343(12)(2020), 112130. 22

work page 2020

[1] [1]

Bialostocki, S

A. Bialostocki, S. Gilboa, Y. Roditty, Anti-Ramsey numbers of small graphs, Ars Combin. 123(2015), 41-53

work page 2015

[2] [2]

Chen, X.L

H. Chen, X.L. Li, J.H. Tu, Complete solution for the rainbow number of match- ings, Discrete Math. 309(10)(2009), 3370-3380

work page 2009

[3] [3]

Erd˝ os, M

P. Erd˝ os, M. Simonovits, V.T. S´ os, Anti-Ramsey theorems, Colloq. Math. Soc. J´ anos Bolyai, Vol. 10 North-Holland Publishing Co., Amsterdam-London, 1975, pp. 633- 643

work page 1975

[4] [4]

C.Q. Fang, E. Gy˝ ori, M. Lu, J.M. Xiao, On the anti-Ramsey number of forests, Discrete Appl. Math. 291(11)(2021), 129-142

work page 2021

[5] [5]

Fujita, A

S. Fujita, A. Kaneko, I. Schiermeyer, K. Suzuki, A rainbow k-matching in the complete graph withrcolors, Electron. J. Combin. 16(2009), #R51. 21

work page 2009

[6] [6]

Gilboa, Y

S. Gilboa, Y. Roditty, Anti-Ramsey numbers of graphs with small connected components, Graphs Combin. 32(2)(2016), 649-662

work page 2016

[7] [7]

R. Haas, M. Young, The anti-Ramsey number of perfect matching, Discrete Math. 312(2012)933-937

work page 2012

[8] [8]

M.L. He, Z.M. Jin, Rainbow short linear forests in edge-colored complete graph, Discrete Appl. Math. 361(2025), 523-536

work page 2025

[9] [9]

Jahanbekam, D.B

S. Jahanbekam, D.B. West, Anti-Ramsey problems for t edge-disjoint rainbow spanning subgraphs: cycles, matchings, or trees, J. Graph Theory 82(1)(2016), 75-89

work page 2016

[10] [10]

Jie, M.L

Q. Jie, M.L. He, Z.M. Jin, Rainbow forest consisting of short paths in Kn, Discrete Appl. Math. 376(2025), 260-269

work page 2025

[11] [11]

Montellano-Ballesteros, V

J.J. Montellano-Ballesteros, V. Neumann-Lara, An anti-Ramsey theorem on cycles, Graphs Combin. 21 (3) (2005), 343-354

work page 2005

[12] [12]

Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math

I. Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157-162

work page 2004

[13] [13]

Simonovits, V.T

M. Simonovits, V.T. S´ os, On restricted coloring of Kn, Combinatorica 4 (1) (1984) 101–110

work page 1984

[14] [14]

Xie, L.T

T.Y. Xie, L.T. Yuan, On the anti-Ramsey numbers of linear forests, Discrete Math. 343(12)(2020), 112130

work page 2020

[15] [15]

Xie, L.T

T.Y. Xie, L.T. Yuan, On the anti-Ramsey numbers of linear forests, Discrete Math. 343(12)(2020), 112130. 22

work page 2020