Rainbow Catalog: Interdisciplinary Perspectives

• Rainbow catalog is a collection of distinct but related notions where maximal color distinctness organizes subgraphs, polygons, simplices, and scattering patterns across disciplines.
• It employs rigorous combinatorial, geometric, and probabilistic methods to establish invariants, thresholds, and reduction techniques for rainbow structures.
• Its applications span from ensuring rainbow connectivity in graphs and optimal constructions in geometry to modeling interference patterns in nuclear scattering.

Searching arXiv for the specified paper and closely related rainbow-geometry references.

“Rainbow catalog” (Editor's term) denotes a family of results in which a color-distinctness condition organizes the target object. In edge-colored graphs, a subgraph is rainbow if all its edges have distinct colours; in colored point sets, a perfect rainbow polygon contains exactly one point of each color; in triangulations of manifolds, a rainbow simplex is a top-dimensional simplex with exactly one vertex in each color class; in random discrete models, the rainbow threshold $p_k(F)$ is defined by the median probability that a randomly colored subset contains a rainbow edge of the minimal family; and in heavy-ion scattering, the nuclear rainbow denotes an Airy structure with superimposed ripple oscillations (Montgomery et al., 2018, Flores-Peñaloza et al., 2020, Montejano, 2018, Han et al., 2023, Ohkubo et al., 2014). The shared vocabulary is therefore not a single theory but a collection of mathematically and physically distinct notions whose common feature is a maximal distinctness or interference pattern.

1. Foundational meanings and parameters

In graph-theoretic usage, the basic object is an edge-colored graph. A subgraph is called rainbow if all its edges have distinct colours, a path is rainbow if no two edges on it are colored the same, and a graph is rainbow-connected if every pair of vertices is joined by a rainbow path; if the rainbow path must also be a shortest path, the graph is strongly rainbow-connected. The associated numerical invariants are the rainbow connection number $\mathrm{rc}(G)$ and the strong rainbow connection number $\mathrm{src}(G)$, with

$$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$

In colored triangle-free graphs, a separate forcing parameter is the minimum color-degree

$\delta^c(H):=\min_{v\in V(H)} d^c(v),$

where $d^c(v)$ counts the number of distinct colors on edges incident to $v$ (Keranen et al., 2014, Czygrinow et al., 12 Jun 2026).

In discrete geometry, the terminology is adapted to colored point sets. For a colored point set $S=\bigcup_{i=1}^k S_i$ in the plane in general position, a rainbow polygon is a simple polygon that contains at most one point of each color, while a perfect rainbow polygon contains exactly one point of each color, either in its interior or on its boundary. The optimization parameter is

$\operatorname{rb\text{-}index}(S),$

the minimum number of vertices in a perfect rainbow polygon for $S$, and its worst-case version

$\mathrm{rc}(G)$0

the maximum of $\mathrm{rc}(G)$1 over all $\mathrm{rc}(G)$2-colored point sets in general position (Flores-Peñaloza et al., 2020).

In topological combinatorics, if $\mathrm{rc}(G)$3 triangulates an $\mathrm{rc}(G)$4-dimensional manifold and its vertices are partitioned into $\mathrm{rc}(G)$5, then a rainbow simplex is an $\mathrm{rc}(G)$6-simplex with exactly one vertex in each color class. The main criterion used there is the homological vanishing condition

$\mathrm{rc}(G)$7

for every nonempty $\mathrm{rc}(G)$8, where $\mathrm{rc}(G)$9 is the induced subcomplex on the union of the color classes indexed by $\mathrm{src}(G)$0 (Montejano, 2018).

A further family of invariants is extremal. For a graph $\mathrm{src}(G)$1, the rainbow Turán number $\mathrm{src}(G)$2 is the largest number of edges in an $\mathrm{src}(G)$3-vertex graph that admits a proper edge-coloring with no rainbow $\mathrm{src}(G)$4 subgraph, while the anti-Ramsey and rainbow numbers satisfy

$\mathrm{src}(G)$5

where $\mathrm{src}(G)$6 is the maximum number of colours in an edge-coloring of $\mathrm{src}(G)$7 with no rainbow copy of $\mathrm{src}(G)$8 (Bednar et al., 2022, Schiermeyer et al., 2012).

Taken together, these definitions show that “rainbow” is not tied to one ambient category. It may denote color injectivity on edges, exact color representation in geometric containment, or a forcing threshold in a random model.

2. Geometric and topological rainbow configurations

For colored point sets in the plane, the central object is the perfect rainbow polygon. The exact small-$\mathrm{src}(G)$9 values determined are

$$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$0

Thus $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$1 is the first case in which the rainbow index is strictly larger than $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$2. For $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$3, the general bounds are

$$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$4

and for a $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$5-colored set of $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$6 points in the plane in general position, a perfect rainbow polygon with at most $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$7 vertices can be computed in $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$8 time (Flores-Peñaloza et al., 2020).

The structural mechanism behind these bounds is an equivalence between rainbow polygons and noncrossing covering trees. The paper reduces the polygon problem to constructing a noncrossing covering tree with small value of $$\mathrm{diam}(G)\le \mathrm{rc}(G)\le \mathrm{src}(G)\le n-1.$$9, where $\delta^c(H):=\min_{v\in V(H)} d^c(v),$0 is the number of pairwise noncrossing segments in a minimum segment decomposition of the tree and $\delta^c(H):=\min_{v\in V(H)} d^c(v),$1 is the sum of multiplicities of the tree’s forks. From such a tree, one can “thicken” it into a simple polygon with $\delta^c(H):=\min_{v\in V(H)} d^c(v),$2 vertices; conversely, a sufficiently thin polygon containing a point set yields such a covering tree. The same reduction supports the lower-bound construction, which leverages hard point sets from covering-path literature and yields the bound $\delta^c(H):=\min_{v\in V(H)} d^c(v),$3 (Flores-Peñaloza et al., 2020).

Rainbow geometry also appears in the existence of equilateral triangles. In $\delta^c(H):=\min_{v\in V(H)} d^c(v),$4 or $\delta^c(H):=\min_{v\in V(H)} d^c(v),$5, if no color class has more than two points and unit equilateral triangles exist, then there must be a rainbow unit equilateral triangle. In the finite-field setting, the quadratic form

$\delta^c(H):=\min_{v\in V(H)} d^c(v),$6

replaces Euclidean distance, and the main asymptotic theorem states that if no color class has size greater than $\delta^c(H):=\min_{v\in V(H)} d^c(v),$7, for any positive constant $\delta^c(H):=\min_{v\in V(H)} d^c(v),$8, then the existence of unit equilateral triangles implies the existence of a rainbow unit equilateral triangle. A sufficient existence condition is that there is $\delta^c(H):=\min_{v\in V(H)} d^c(v),$9 with $d^c(v)$0 (Senger, 2017).

For triangulations of manifolds, the rainbow-simplex problem is controlled by homology. The general criterion is that if

$d^c(v)$1

for every nonempty $d^c(v)$2, then $d^c(v)$3 contains a rainbow simplex. The manifold-specific theorems specialize this principle to dimensions $d^c(v)$4, $d^c(v)$5, and $d^c(v)$6, and to general closed $d^c(v)$7-manifolds via derived neighborhoods and strong deformation retracts. In the sphere case, the result is recovered via Alexander duality (Montejano, 2018).

These geometric and topological instances share a common formal pattern: color classes are distributed over points or vertices, and rainbow existence means exact representation of all colors in a minimally structured object. This suggests a unifying catalog-level distinction between containment problems, such as rainbow polygons and simplices, and packing or threshold problems, which arise in graph theory.

3. Rainbow structures in edge-colored graphs

A large part of the literature concerns spanning or nearly spanning rainbow structures in properly colored graphs. In properly colored $d^c(v)$8, if at most $d^c(v)$9 colours have more than $v$0 edges, then there are $v$1 edge-disjoint perfect rainbow matchings. Under analogous assumptions for $v$2, if at most $v$3 colours have more than $v$4 edges, then there are $v$5 edge-disjoint rainbow Hamiltonian cycles. Most notably, every properly coloured $v$6 has $v$7 edge-disjoint spanning rainbow trees, yielding an asymptotic form of the Brualdi–Hollingsworth and Kaneko–Kano–Suzuki conjectures (Montgomery et al., 2018).

A complementary embedding theorem works in locally $v$8-bounded colorings. If $v$9, then any locally $S=\bigcup_{i=1}^k S_i$0-bounded edge-coloring of $S=\bigcup_{i=1}^k S_i$1 contains a rainbow copy of every tree with at most $S=\bigcup_{i=1}^k S_i$2 vertices. Since such a coloring may have only $S=\bigcup_{i=1}^k S_i$3 distinct colours, this is essentially tight. The proof decomposes the target tree into a chain $S=\bigcup_{i=1}^k S_i$4 and combines greedy embedding, deterministic switching arguments for large stars, probabilistic handling of many leaf-attachments, and a random-vertex/random-colour path-finding lemma for bare paths of length $S=\bigcup_{i=1}^k S_i$5 (Montgomery et al., 2018).

Long rainbow cycles are forced even without a full decomposition theorem. Every properly edge-colored complete graph on $S=\bigcup_{i=1}^k S_i$6 vertices contains a rainbow cycle on at least $S=\bigcup_{i=1}^k S_i$7 vertices, improving the earlier $S=\bigcup_{i=1}^k S_i$8 bound. The same paper proves that every properly colored complete graph has a Hamilton cycle using at least $S=\bigcup_{i=1}^k S_i$9 different colors. The first result uses random color-class sampling, nearly-rainbow bipartitions, and expansion-based path absorption; the second uses swap-optimal spanning rainbow path forests and a nested family argument modeled on Hatami–Shor (Balogh et al., 2017).

For triangle-free graphs, color-degree thresholds force short and long even rainbow cycles. If $\operatorname{rb\text{-}index}(S),$0 is an edge-colored triangle-free graph on sufficiently large $\operatorname{rb\text{-}index}(S),$1 vertices and

$\operatorname{rb\text{-}index}(S),$2

then $\operatorname{rb\text{-}index}(S),$3 contains a rainbow cycle of length four, and this bound is best possible. More generally, for every fixed $\operatorname{rb\text{-}index}(S),$4, there is $\operatorname{rb\text{-}index}(S),$5 such that for $\operatorname{rb\text{-}index}(S),$6, if

$\operatorname{rb\text{-}index}(S),$7

then $\operatorname{rb\text{-}index}(S),$8 contains a rainbow $\operatorname{rb\text{-}index}(S),$9. The proof uses an associated digraph $S$0, regularity, a reduced digraph, and a stability analysis around a 5-partite cyclic extremal structure (Czygrinow et al., 12 Jun 2026).

This body of work establishes rainbow subgraphs as a full-scale packing, embedding, and forcing theory rather than a sequence of isolated anti-Ramsey questions.

4. Extremal, anti-Ramsey, and connectivity theories

Rainbow connectivity studies color assignments that enforce rainbow paths between vertices. For every $S$1, deciding whether $S$2 is NP-complete for split graphs, and there is no polynomial-time approximation algorithm for $S$3 on $S$4-vertex split graphs with approximation factor $S$5 for any $S$6 unless $S$7. On block graphs, by contrast, if $S$8 denotes the number of blocks containing fewer than $S$9 cut vertices, then

$\mathrm{rc}(G)$00

and $\mathrm{rc}(G)$01 can be computed in $\mathrm{rc}(G)$02 time. The same paper also gives a polynomial-time characterization of bridgeless block graphs with $\mathrm{rc}(G)$03 (Keranen et al., 2014).

Anti-Ramsey theory replaces connectivity by unavoidable rainbow copies in edge-colored complete graphs. For connected graphs $\mathrm{rc}(G)$04 of order $\mathrm{rc}(G)$05 with cyclomatic number $\mathrm{rc}(G)$06, $\mathrm{rc}(G)$07 cannot be bounded above by any function linear in $\mathrm{rc}(G)$08. By contrast, if $\mathrm{rc}(G)$09, then the rainbow number is linearly bounded in $\mathrm{rc}(G)$10. The paper computes exact or explicit values for several small graphs: for the bull $\mathrm{rc}(G)$11,

$\mathrm{rc}(G)$12

for the diamond $\mathrm{rc}(G)$13, exact values are given for $\mathrm{rc}(G)$14, including

$\mathrm{rc}(G)$15

and further exact values and bounds are established for $\mathrm{rc}(G)$16 and the house graph $\mathrm{rc}(G)$17 (Schiermeyer et al., 2012).

A related extremal parameter is the rainbow Turán number $\mathrm{rc}(G)$18, the maximum number of edges in an $\mathrm{rc}(G)$19-vertex graph that can be properly edge-colored with no rainbow $\mathrm{rc}(G)$20 subgraph. The reduction method developed for trees augments a target tree $\mathrm{rc}(G)$21 to a graph $\mathrm{rc}(G)$22 such that every proper edge-coloring of $\mathrm{rc}(G)$23 contains a rainbow $\mathrm{rc}(G)$24, after which classical Turán bounds on $\mathrm{rc}(G)$25 yield upper bounds on $\mathrm{rc}(G)$26. This method gives results for double stars, caterpillars, perfect binary trees, and perfect $\mathrm{rc}(G)$27-ary trees. The same work also introduces $\mathrm{rc}(G)$28-unique colorings and $\mathrm{rc}(G)$29-unique Turán numbers $\mathrm{rc}(G)$30 (Bednar et al., 2022).

These theories formalize several different senses in which rainbow structure can be forced: by optimizing over colorings, by maximizing edges under rainbow avoidance, or by asking for colorings that guarantee pairwise connectivity.

5. Random models and threshold theorems

The threshold theory of rainbow structures places coloring inside the random model itself. Let $\mathrm{rc}(G)$31 be a finite ground set and $\mathrm{rc}(G)$32 a number of colors. The model first includes each $\mathrm{rc}(G)$33 independently with probability $\mathrm{rc}(G)$34, then independently assigns a uniform random color from $\mathrm{rc}(G)$35 to each included element. Equivalently, one works on $\mathrm{rc}(G)$36. If $\mathrm{rc}(G)$37 is increasing and $\mathrm{rc}(G)$38 is the family of minimal elements of $\mathrm{rc}(G)$39, then $\mathrm{rc}(G)$40 is the increasing family generated by all rainbow edges of $\mathrm{rc}(G)$41, and the rainbow threshold $\mathrm{rc}(G)$42 is defined by

$\mathrm{rc}(G)$43

The central theorem states that there exists an absolute constant $\mathrm{rc}(G)$44 such that for every finite $\mathrm{rc}(G)$45, every increasing $\mathrm{rc}(G)$46, and every $\mathrm{rc}(G)$47,

$\mathrm{rc}(G)$48

where $\mathrm{rc}(G)$49 is the fractional expectation-threshold (Han et al., 2023).

The spread-theoretic formulation is equally important. If $\mathrm{rc}(G)$50 is an $\mathrm{rc}(G)$51-bounded $\mathrm{rc}(G)$52-spread multi-hypergraph and the vertices are colored independently and uniformly from $\mathrm{rc}(G)$53, with $\mathrm{rc}(G)$54, then for

$\mathrm{rc}(G)$55

the random set $\mathrm{rc}(G)$56 contains a rainbow edge of $\mathrm{rc}(G)$57 with probability $\mathrm{rc}(G)$58. A stronger rainbow analogue of Spiro’s theorem is also proved for $\mathrm{rc}(G)$59-spread hypergraphs. The key technical device is a transversal model on $\mathrm{rc}(G)$60 together with a McDiarmid-style coupling lemma showing that success in the transversal model transfers to the rainbow-colored model (Han et al., 2023).

The applications show that rainbow thresholds track the uncolored thresholds to the same order in several prominent settings. For fixed $\mathrm{rc}(G)$61 with $\mathrm{rc}(G)$62, if the edges of $\mathrm{rc}(G)$63 are independently colored with more than $\mathrm{rc}(G)$64 colors and

$\mathrm{rc}(G)$65

then with probability tending to $\mathrm{rc}(G)$66 the random hypergraph contains a rainbow Hamilton $\mathrm{rc}(G)$67-cycle. For every $\mathrm{rc}(G)$68, if the edges of $\mathrm{rc}(G)$69 are colored with more than $\mathrm{rc}(G)$70 colors and

$\mathrm{rc}(G)$71

then the graph contains a rainbow $\mathrm{rc}(G)$72-th power of a Hamilton cycle with probability at least $\mathrm{rc}(G)$73. Further results cover rainbow bounded-degree spanning trees in Dirac graphs and rainbow perfect matchings in dense hypergraphs (Han et al., 2023).

This threshold theory reframes rainbow containment as a probabilistic forcing problem and links it to the expectation-threshold and spreadness framework developed for uncolored random structures.

6. Nuclear rainbow and ripple structure

Outside combinatorics, the term “rainbow” also appears in scattering theory. In $\mathrm{rc}(G)$74 elastic scattering at incident energy $\mathrm{rc}(G)$75 MeV, the nuclear rainbow is the broad Airy structure produced by strong refraction in the nuclear mean field, and the paper identifies finer oscillations superimposed on that Airy structure as nuclear ripples. The analysis uses a coupled channels calculation with an extended double-folding model based on a density-dependent effective nucleon-nucleon force and realistic microscopic wave functions for $\mathrm{rc}(G)$76C and $\mathrm{rc}(G)$77O, with folding potential

$\mathrm{rc}(G)$78

The coupled channels treatment includes the $\mathrm{rc}(G)$79, $\mathrm{rc}(G)$80, and $\mathrm{rc}(G)$81 states of $\mathrm{rc}(G)$82C and the $\mathrm{rc}(G)$83 and $\mathrm{rc}(G)$84 states of $\mathrm{rc}(G)$85O (Ohkubo et al., 2014).

The main physical claim is that the ripples arise from interference between refractive farside waves and reflective nearside waves, in direct analogy with the interference between refracted and externally reflected waves in the meteorological rainbow. A single-channel calculation with a smooth imaginary potential reproduces the gross Airy pattern but misses the oscillations; the ripples appear either with an extremely thin-skinned imaginary potential or, more realistically, when channel coupling to excited states is included. In that interpretation, coupling to excited states generates the external reflection needed for ripples. Partial-wave analysis locates the onset of the effect around $\mathrm{rc}(G)$86 for the backward Airy peak $\mathrm{rc}(G)$87 and $\mathrm{rc}(G)$88 for the intermediate-angle peak $\mathrm{rc}(G)$89, corresponding to impact parameters $\mathrm{rc}(G)$90–$\mathrm{rc}(G)$91 fm (Ohkubo et al., 2014).

This usage is formally distinct from the graph-theoretic and geometric meanings of “rainbow,” since it concerns wave interference rather than color classes. A plausible implication is that the term functions across disciplines as a descriptor of structured multiplicity: either all colors are represented distinctly, or multiple wave components interfere to produce a rainbow-like pattern.