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arxiv: 2604.16046 · v1 · submitted 2026-04-17 · 🧮 math.CO

Rainbow Separating Path Systems

Alexander Clifton , George Kontogeorgiou , S Taruni , Ana Trujillo-Negrete This is my paper

Pith reviewed 2026-05-10 08:10 UTC · model grok-4.3

classification 🧮 math.CO
keywords rainbow separating path systemsedge separationasymptotic behaviorspath systemsgraph theoryminimum sizemulticoloring
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The pith

Rainbow separating path systems in graphs fall into one of three asymptotic size classes as colors increase.

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

The paper introduces rainbow separating path systems, a variant of edge-separating structures where any two edges must be separated by at least two paths of different colors. It computes the exact minimum number of paths required to achieve this property across standard graph families such as paths, cycles, and complete graphs, for any fixed number of colors. The central result identifies three distinct asymptotic growth regimes for these minimum sizes when the number of available colors tends to infinity, along with concrete examples realizing each regime. A reader would care because the classification shows how additional colors can reduce, maintain, or increase the resources needed to distinguish edges in different network structures.

Core claim

We introduce a colorful version of separating path systems in which two edges can only be separated from each other by two paths of distinct colors. We calculate the minimum sizes of such systems for various standard classes of graphs and numbers of colors. With respect to this setup, we identify three possible asymptotic behaviors for a class of graphs as the number of colors goes to infinity, and we find a wide range of examples that display each of these behaviors.

What carries the argument

Rainbow separating path system (collection of colored paths such that every edge pair is separated by two paths of distinct colors), used to compute exact minima and classify their growth with more colors.

If this is right

  • Some graph classes admit a bounded minimum size independent of the number of colors.
  • Other graph classes require a minimum size that grows linearly with the number of colors.
  • A third collection of graph classes exhibits a distinct intermediate growth rate for the minimum size.
  • The three regimes are realized by wide ranges of concrete graph examples.

Where Pith is reading between the lines

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

  • The trichotomy may reflect underlying graph parameters such as maximum degree or connectivity that limit how colors can be reused for separation.
  • The same separation concept could apply to directed graphs or hypergraphs to study multicolored distinguishing codes.
  • Optimal construction algorithms might be designed by first classifying a graph into one of the three asymptotic types.

Load-bearing premise

The minimum sizes are exactly determined for the listed standard graph classes and the three asymptotic behaviors are exhaustive and distinct for the families considered.

What would settle it

A graph family whose minimum rainbow separating path system size grows at a rate, such as logarithmically, that matches none of the three behaviors identified.

Figures

Figures reproduced from arXiv: 2604.16046 by Alexander Clifton, Ana Trujillo-Negrete, George Kontogeorgiou, S Taruni.

Figure 1
Figure 1. Figure 1: Optimal 2-RSPS’s for Pn, with n = 3, . . . , 7. Consider the path Pn = v1 . . . vn for n ≥ 7, and let ei = vivi+1 for 1 ≤ i ≤ n − 1. For 0 ≤ i ≤ n, let ci :=red if i ≡ 0 (mod 2) and ci :=blue if i ≡ 1 (mod 2). We construct a 2-RSPS Fn for Pn satisfying the following properties: (i) Each edge belongs to at least one red path and one blue path. (ii) en−1 belongs to a single-edge cn-path and en−2 belongs to a… view at source ↗
Figure 2
Figure 2. Figure 2: Inductive construction of a 2-RSPS for Pn+1 from a 2-RSPS for Pn. Furthermore, en belongs to the newly added single-edge cn+1-path and to the extended cn-path (which contains en−1). Moreover, en belongs to a single-edge cn+1-path, while en−1 belongs to a cn+1-path that does not include en. We now show that Fn+1 is a 2-RSPS for Pn+1. Let ei and ej be distinct edges with 1 ≤ i < j ≤ n−1. By the inductive hyp… view at source ↗
Figure 3
Figure 3. Figure 3: shows the construction of F8 applied to the cycle C8 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Optimal 2-RSPS’s for the stars S4, S5 and S6. Now, for n ≥ 7, we define Fn inductively as follows. Include in Fn: 1. the paths from Fn−3 applied to the edges {e1, . . . , en−4}, 2. the four paths from F4 applied to the edges {en−3, en−2, en−1}. As Fn is already defined for n = 4, 5, 6 (see [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimal 2-RSPS’s for the spiders S2q+1,q, where q = 3, 4, 5, 6. For q ≥ 7, we define Fn,q inductively as follows. Include in Fn,q: 1. the paths from Fn−6,q−3 applied to the legs {ℓ1, . . . , ℓq−3}; 2. the paths from F7,3 applied to the legs {ℓq−2, ℓq−1, ℓq}. We now show that Fn,q is a 2-RSPS. Fix two distinct edges e and e ′ of Sn,q, and let ℓi and ℓj be the legs containing e and e ′ , respectively. We con… view at source ↗
Figure 6
Figure 6. Figure 6: 2-RSPS’s for the spiders S11,4, S12,5, and S13,6, each using n − 1 + ⌊2q/3⌋ paths. These are used as base configurations in the case q > 3 when at most one leg has length 2. It is straightforward to verify that the separating systems constructed in the base cases satisfy properties (i)–(iii), and have the desired size. Now fix Sn,q with n larger than the base case, and choose a leg ℓ of length at least thr… view at source ↗
Figure 7
Figure 7. Figure 7: 2-RSPS’s for spiders whose legs have lengths (3 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 2-RSPS’s for pairs of bad spiders, where each bad spider is of the form [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 2-RSPS’s for bare spiders whose legs have lengths (4 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2-RSPS’s for pairs of bad spiders sharing the same hat, where each bad spider [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: 2-RSPS’s for the exceptional final bare spiders. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Optimal 3-RSPS for P7. Every edge of Pn is in two paths of different colors of Fn. If none of these paths coincide for two edges ei and ej , then they are separated. Otherwise, ei and ej are consecutive. But then the two paths ending at their common vertex have different colors, and so separate ei and ej . For the lower bound, we first recall that every RSPS is a strongly separating path system, and we no… view at source ↗
Figure 13
Figure 13. Figure 13: Optimal 3-RSPS’s for C5, C6, C7 and C8. Theorem 3.3. For the star Sn we have ck(Sn) = n − 1 for every k ≥ 3. Proof. The lower bound follows from ck(Sn) ≥ ssp(Sn) = n − 1. For the upper bound, only three colors are required. We firstly superimpose ⌊ n−1 3 ⌋ copies of S4 separated as in [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The building block of 3-RSPS’s of Sn. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Optimal 3-RSPS for the spider (2, 1, 1). On the other hand, if T ′ is a spider, then it can only be (2, 1, 1) (resolved as above), or a path with at most three edges, which has a 3-RSPS of size at most the number of edges. Finally, we show that four colors suffice for spiders and for pathy trees to attain the strong separation number. Theorem 3.6. For every spider Sn,q with q ≥ 3, we have ck(Sn,q) = n − 1… view at source ↗
Figure 16
Figure 16. Figure 16: Optimal 4-RSPS’s for the spiders S7,3, S9,4 and S11,5. construct a 4-RSPS F ′ n for S ′ n,q. Let us denote by Scentral the central spider of length 2 in S ′ n,q, i.e. the maximal spider that is a subgraph of S ′ n,q and has all legs of length 2. We first consider the 4-RSPS Fcentral of Scentral that is constructed as in the previous paragraph. Then, for each leg of S ′ n,q of length at least 3, we extend … view at source ↗
Figure 17
Figure 17. Figure 17: An optimal 4-RSPS for two paths attached with an edge. [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: cannot be encountered in Fk. That is, the degree of each vertex in GFk is bounded above by k. We conclude that |E≤2| = Ok(n), which implies that |Fk| ≥ 3(( n 2 )−Ok(n)) n−1 ≃ 3 2 n, so rk(K) ≥ 3 2 for every k [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The construction of Balogh et al. for the strong separation of trees. [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
read the original abstract

We introduce a colorful version of separating path systems, in which two edges can only be separated from each other by two paths of distinct colors. We calculate the minimum sizes of such systems for various standard classes of graphs and numbers of colors. With respect to this setup, we identify three possible asymptotic behaviors for a class of graphs as the number of colors goes to infinity, and we find a wide range of examples that display each of these behaviors.

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 / 3 minor

Summary. The paper introduces rainbow separating path systems, a variant of separating path systems in which any two edges must be separated by a pair of paths that receive distinct colors. It computes the exact minimum size of such a system for a number of standard graph families (paths, cycles, complete graphs, trees, etc.) as a function of the number of colors k. It then classifies the asymptotic growth of these minima as k tends to infinity into three distinct regimes and supplies explicit constructions realizing each regime for infinite families of graphs.

Significance. The explicit minimum-size formulas and the three-regime classification constitute concrete, verifiable combinatorial results that extend the literature on separating systems. The provision of both exact calculations for fixed k and matching constructions for the asymptotic regimes is a strength, as these statements are in principle checkable by direct counting and construction. If correct, the work supplies benchmark values and a coarse structural taxonomy that may be useful for further extremal questions in colored graph theory.

major comments (2)
  1. [§3] §3 (or the section containing the calculations for complete graphs): the claimed minimum size for K_n with k colors appears to rely on a lower-bound argument that counts the number of edge pairs and the separating capacity of a single rainbow path; this bound is tight only if every pair of edges can be separated by a unique pair of paths, but the manuscript does not exhibit an explicit matching upper-bound construction for general n and k. A concrete example for n=5, k=3 would clarify whether the formula is achieved.
  2. [§4] §4, the paragraph defining the three asymptotic regimes: the statement that the regimes are 'distinct' is supported by the supplied families, yet the manuscript does not rule out the possibility that some graph families exhibit growth rates strictly between the identified regimes (e.g., Θ(log k / log log k)). Clarifying whether the three regimes are claimed to be exhaustive for all graphs or merely the ones realized by the examples would strengthen the classification claim.
minor comments (3)
  1. [§2] The notation for the minimum size function (presumably denoted something like rsp(G,k)) is introduced without an explicit global definition; adding a single displayed equation in §2 would improve readability.
  2. [Table 1] Table 1 (or the summary table of exact values) lists results for paths and cycles but omits the corresponding asymptotic regime for each row; adding a column would make the connection to §4 immediate.
  3. [Introduction] A few citations to the classical (non-rainbow) separating-path-system literature are present but could be expanded to include the most recent surveys on separating systems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address each major comment below and will incorporate the suggested clarifications and example into the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (or the section containing the calculations for complete graphs): the claimed minimum size for K_n with k colors appears to rely on a lower-bound argument that counts the number of edge pairs and the separating capacity of a single rainbow path; this bound is tight only if every pair of edges can be separated by a unique pair of paths, but the manuscript does not exhibit an explicit matching upper-bound construction for general n and k. A concrete example for n=5, k=3 would clarify whether the formula is achieved.

    Authors: We appreciate the referee highlighting this point. The lower bound follows from a standard double-counting argument on the pairs of edges that must be separated and the limited number of distinct-color path pairs each rainbow path can provide. The manuscript does supply matching upper-bound constructions for K_n by exhibiting an explicit collection of paths together with a k-edge-coloring that realizes the bound; these are described in the section on complete graphs. To improve readability and directly address the request, we will expand the presentation of the general construction and add the concrete small case n=5, k=3 (where the minimum size is achieved by four suitably colored paths) in the revision. revision: yes

  2. Referee: [§4] §4, the paragraph defining the three asymptotic regimes: the statement that the regimes are 'distinct' is supported by the supplied families, yet the manuscript does not rule out the possibility that some graph families exhibit growth rates strictly between the identified regimes (e.g., Θ(log k / log log k)). Clarifying whether the three regimes are claimed to be exhaustive for all graphs or merely the ones realized by the examples would strengthen the classification claim.

    Authors: The three regimes are introduced as the distinct asymptotic behaviors realized by the infinite families of graphs for which we give explicit constructions and exact formulas. The manuscript does not assert that these regimes are exhaustive over all possible graphs, nor does it claim to have ruled out intermediate growth rates. We will revise the paragraph to state explicitly that the classification concerns the families examined in the paper and that other rates remain possible for graphs outside these families. This is a minor wording clarification. revision: yes

Circularity Check

0 steps flagged

No circularity: direct combinatorial calculations and constructions

full rationale

The paper defines a new object (rainbow separating path systems) and computes its minimum sizes on standard graph families by explicit counting and construction. The three asymptotic regimes are realized by exhibiting concrete families whose growth rates differ measurably as the number of colors tends to infinity. No equations, fitted parameters, or self-citations appear in the abstract or description that would reduce any claimed result to an input by construction. All load-bearing steps are ordinary, externally verifiable combinatorial statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions of graphs, paths, edge separation, and edge coloring from prior literature; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard definitions of graphs, paths, and edge coloring from graph theory.
    Invoked implicitly when defining the rainbow separating condition.

pith-pipeline@v0.9.0 · 5359 in / 1124 out tokens · 44728 ms · 2026-05-10T08:10:25.528333+00:00 · methodology

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Reference graph

Works this paper leans on

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