Rainbow Linear Forests in Graph Theory

• Rainbow linear forests are graph structures composed of vertex-disjoint paths with uniquely colored edges, playing a pivotal role in combinatorial optimization and anti-Ramsey theory.
• They underpin fundamental results in anti-Ramsey theory by establishing sharp bounds on the maximum number of colors avoiding rainbow subgraphs in complete graphs.
• Algorithmic studies reveal a complex landscape, with problems ranging from APX-complete cases to polynomial-time solvable instances, emphasizing the problem's computational depth.

A rainbow linear forest is a graph-theoretic structure arising in the study of edge-colorings and extremal combinatorics, characterized by the union of vertex-disjoint paths with each edge assigned a distinct color under a rainbow constraint. The concept plays a central role in combinatorial optimization, anti-Ramsey theory, and complexity analysis of rainbow subgraphs, connecting questions about colorings of complete graphs to hardness and algorithmic feasibility in subgraph detection.

1. Formal Definition of Rainbow Linear Forests

Let $G=(V,E)$ be a finite undirected graph equipped with an edge-coloring $\varphi:E\to C$. All components of a linear forest are paths: explicitly, a linear forest is a forest each of whose components is a path $P_\ell$ (a chordless graph on $\ell$ vertices and $\ell-1$ edges). For a subgraph $H\subseteq G$, $H$ is called a rainbow subgraph if all its edges are assigned distinct colors; i.e., for $e\neq f\in E(H)$, $\varphi(e)\neq\varphi(f)$. A rainbow linear forest is a rainbow subgraph in which every connected component is a path.

In the generalized context of multigraph collections $G=(G_1,\ldots,G_m)$ on a common vertex set $V$ (Li et al., 13 Dec 2025), a rainbow linear forest $H$ is one with edges $e\in E(H)$ such that there exists an injective mapping $c:E(H)\to[m]$ and $e\in E(G_{c(e)})$ for all $e$.

2. Anti-Ramsey Theory and Extremal Results

The anti-Ramsey number $AR(n,F)$ for a graph $F$ is the maximal number of colors assigned to $E(K_n)$ so that there is no rainbow copy of $F$ in $K_n$. For linear forests $F=\bigcup_{i=1}^k P_{t_i}$, Xie and Yuan established an exact formula for all sufficiently large $n$ when $F$ contains at least one even path (Xie et al., 2020), showing that:

$$AR(n,F) = \binom{S-2}{2} + (S-2)(n-(S-2)) + 1 + \varepsilon,$$

where $S=\sum_{i=1}^k\lfloor t_i/2\rfloor$, and $\varepsilon=1$ if exactly one $t_i$ is even, $\varepsilon=0$ otherwise. Extremal colorings achieving this bound are constructed by choosing a set $U\subset V(K_n)$ of size $S-2$, coloring every edge incident to $U$ with a distinct color, and coloring the remaining edges on $T=V\setminus U$ with only $1+\varepsilon$ colors, precluding rainbow copies of $F$.

For spanning linear forests composed of $kP_3\cup tP_2$ on $n=3k+2t$ vertices, Ghalavand–Li give (Ghalavand et al., 30 Sep 2025) the sharp bound

$AR(n,kP_3\cup tP_2) = \frac12(n-3)(n-4)+1,$

with constructions based on a rainbow $K_{n-3}$ and a single color for the edges connecting the three remaining vertices.

Approximate upper bounds for arbitrary linear forests are given by: $ar(K_n, F) = (\nu(F)-1)n + O(1),$ where $\nu(F)$ is the matching number of $F$ (Fang et al., 2019). For small linear forests, exact values are established for cases such as $F=2P_4$ and $F=kP_3$.

3. Computational Complexity and Algorithmics

The maximum rainbow matching problem for an edge-colored graph $G$ seeks $$\mathrm{RLM}(G) = \max\{ |M| : M \text{ is a rainbow matching in } G\}$$, where a matching is a set of pairwise non-incident edges. The complexity landscape for rainbow matching in linear forests is characterized by pronounced dichotomy (Le et al., 2013):

• APX-completeness: MAX RAINBOW MATCHING remains APX-complete even for properly edge-colored linear forests that are $P_5$-free. No polynomial time algorithm attains an approximation ratio better than $\frac{423}{424}$ unless P=NP.
• Polynomial-time solvability: For $P_4$-free linear forests, MAX RAINBOW MATCHING is computable in $O(m^{3/2})$ via reduction to bipartite matching.
• Parameterized complexity: The problem is FPT in the matching size $k$ for $P_5$-free forests, with $O(n+2^k k^3)$ time in the $k$-forest variant.

The reduction to APX-completeness is via intricate constructions involving color-line graphs and faithfully encoding instances of 3-MIS in triangle-free 3-regular graphs.

A $2/3-\varepsilon$ approximation for arbitrary graphs is obtainable by leveraging the property that their color-line graphs are $K_{1,4}$-free, using Hurkens–Schrijver’s result for $K_{1,4}$-free graphs.

4. Embedding Rainbow Linear Forests in Hamiltonian Structures

Recent work extends the classical Ore’s Theorem to the rainbow setting, embedding prescribed rainbow linear forests into rainbow Hamiltonian paths under degree-sum constraints (Li et al., 13 Dec 2025). Specifically, in a multigraph system $G=(G_1,\ldots,G_n)$ on $n$ vertices, if for every $i$ and every non-edge $uv\notin E(G_i)$ one has $d_{G_i}(u)+d_{G_i}(v)\ge n$, then either every pair is connected by a rainbow Hamiltonian path containing the linear forest $H$ (with up to $k\leq (n-4)/3$ edges embedded), or the system falls into narrowly described extremal structures.

The proof employs inductive absorption, stepwise integration of the forest components into the path by Ore-type rotations, and a dichotomy into "good path case" or rigidly partitioned exceptions, fully characterizing the embeddability threshold for rainbow linear forests in Hamiltonian contexts.

5. Rainbow Commonness and Forest Structure

The property of $r$-rainbow commonness asks whether the maximum number of rainbow copies of a graph $H$ in an $r$-edge coloring of $K_n$ occurs asymptotically in the uniform random coloring. De Silva et al. conjectured that paths $P_s$ are $s$-rainbow common and proved that unions of stars are rainbow common (Sun, 2023). Xie–Yuan establish that any graph with a cycle is $r$-rainbow uncommon for all $r\geq e(H)$, confirming that only forests can be rainbow common. For linear forests, this reduces the question to verifying certain Sidorenko-type inequalities on color-density graphons. Full classification of which rainbow linear forests are $r$-rainbow common remains an active research problem.

6. Structural Extremality and Coloring Constructions

Extremal constructions for anti-Ramsey numbers of linear forests exploit combinatorial partitioning, minimum-degree bootstrapping, coloring of large cliques or bipartite graphs, and tight representing-subgraph analysis. For example, a coloring on a large clique plus one color for edges outside yields optimal solutions up to an additive constant. The partition into cases based on parity and matching number governs the transition between clique-based and bipartite-based extremal colorings (Fang et al., 2019).

These constructions also manifest in complexity reductions and help discern the boundaries between tractable and intractable instances in rainbow matching search and anti-Ramsey optimization.

7. Open Problems and Future Directions

Outstanding questions include the full classification of rainbow commonness for general linear forests, precise determination of anti-Ramsey numbers for arbitrary trees and "spiders" (centered trees with leg length at least two), and tighter embedding thresholds for rainbow linear forests in structures beyond Hamiltonian paths. Parameterized tractability for $P_6$-free and $P_7$-free forests remains unresolved (Le et al., 2013). Extensions to robust rainbow embedding under relaxed degree-sum constraints and deeper integration with Turán-type extremal graph theory are ongoing.

Taken together, rainbow linear forests serve as a rich testing ground for the interplay of color-constraints, combinatorial optimization, and graph-theoretic structure, linking algorithmic complexity and extremal coloring theory in modern discrete mathematics.