Rainbow Hamiltonian Paths: Extremal & Spectral Insights

• Rainbow Hamiltonian paths are spanning paths in colored graph systems where each edge comes from a distinct color class, ensuring unique edge selections.
• The study establishes sharp edge-count and spectral radius thresholds that guarantee rainbow Hamiltonian cycles using methods like the Kelmans transform.
• Deterministic and probabilistic techniques are integrated to characterize extremal configurations and open avenues for further combinatorial research.

A rainbow Hamiltonian path is a topic at the intersection of extremal graph theory, randomized combinatorics, and spectral graph theory. It extends classic Hamiltonian notions to multigraph systems equipped with a “color” or indexing structure, demanding that each edge of the Hamiltonian path or cycle arise distinctly from a prescribed family of graphs or color classes. The study incorporates both deterministic and probabilistic techniques, with sharp structural, spectral, and combinatorial results delineating thresholds and obstructions. Below is a rigorous encyclopedic account of the area, centered on recent breakthroughs, particularly spectral and edge-count extremal conditions (Zhang et al., 31 Jan 2024).

1. Definition and Fundamental Concepts

Let $V$ be an $n$-element set, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a family of $n$ undirected graphs, each with vertex set $V$. Interpreting $G_i$ as "color $i$," a Hamiltonian cycle $C$ of the complete graph on $V$ is called a rainbow Hamiltonian cycle (with respect to $\mathcal{G}$) if its $n$ edges $e_1, \ldots, e_n$ satisfy $e_i \in E(G_i)$ for each $i$; that is, no two edges of $C$ come from the same $G_i$, and altogether $C$ visits each vertex exactly once (Zhang et al., 31 Jan 2024).

Formally, for a general graph system, a rainbow Hamiltonian path is a spanning path whose edges are drawn from distinct color classes—so each is taken from a distinct $G_i$—and similarly for a cycle. This definition accommodates both set systems and edge-colored models, making it applicable to a wide variety of classical and modern combinatorial problems.

2. Extremal Edge-Count and Spectral Radius Theorems

Two main quantitative thresholds guarantee the existence of rainbow Hamiltonian cycles in such systems: the Ore-size (edge-count) condition and the spectral radius criterion (Zhang et al., 31 Jan 2024).

2.1. Edge-Count Theorem

Theorem 2.1 ("rainbow Ore size"): For $n\geq 4$, if $\mathcal{G} = \{G_1, \ldots, G_n\}$ are graphs on a common vertex set $V$, and

$$e(G_i) > \binom{n-1}{2} + 1 \quad \text{ for all } i = 1, \ldots, n,$$

then $\mathcal{G}$ admits a rainbow Hamiltonian cycle (Zhang et al., 31 Jan 2024).

The proof employs:

• The Kelmans transform, shifting each $G_i$ to a threshold graph $K(G_i)$ with nested neighborhoods, ensuring no decrease in edge count.
• A forcing argument: edges with specific symmetry (so-called "mirror edges") must appear in all $K(G_i)$, or else contradict the edgecount minimum.
• A contradiction arises by explicit assembly using these mirror and bridge edges—if a rainbow Hamiltonian cycle cannot be formed, then the edge count of any $K(G_i)$ must in fact not exceed the threshold, violating the initial premise.

2.2. Spectral Radius Criterion

Theorem 3.1 ("rainbow Fiedler–Nikiforov"): For $n\geq 4$, if

$$\rho(G_i) > n-2 \text{ for each } G_i \in \mathcal{G},$$

where $\rho(G)$ is the spectral radius of the adjacency matrix, then $\mathcal{G}$ admits a rainbow Hamiltonian cycle unless all $G_i$ coincide with the extremal threshold graph $K_1 \vee (K_{n-2} \cup K_1)$.

Key steps:

• Stanley’s bound: $e(G) \geq \rho(G)(\rho(G)+1)/2$ ensures edge-count exceeds the Ore-size threshold when $\rho(G) > n-2$.
• Spectral lemmas: Only $K_1 \vee (K_{n-2} \cup K_1)$ (and for $n=5$ an exceptional $K_2 \vee 3 K_1$) can have spectral radius exceeding $n-2$ without being Hamiltonian.
• Replication of the edge-count proof structure: contradictions emerge by replacing edge-count with spectral radius in the critical steps.

3. Extremal Structures and Sharpness

A comprehensive classification of equality (extremal) cases is provided for both the edge-count and spectral criteria:

• The only obstruction under the edge-count condition $e(G_i) \geq \binom{n-1}{2} + 1$ is when all $G_i$ are isomorphic to $K_1 \vee (K_{n-2} \cup K_1)$.
• For the spectral radius criterion, the same extremal configuration is seen, with $K_2 \vee 3K_1$ arising as an exception for $n=5$ but not satisfying the spectral threshold.

In these cases, the color classes "trap" the necessary connectivity, precluding the assembly of a full rainbow Hamiltonian cycle even though the marginal constraints are tight (Zhang et al., 31 Jan 2024).

4. Proof Methods and Combinatorial Techniques

The robust proof framework consists of transformation and structural forcing steps:

• Kelmans Transformation and Threshold Graphs: The argument begins with a shifting procedure ensuring each $G_i$ becomes a threshold graph (i.e., has nested neighborhoods), which are tractable extremal objects for degree/edge analysis.
• Mirror Edge and Neighborhood Forcing: By numbering vertices and analyzing "mirror edges," one forces wide edge overlap across all $K(G_i)$ via combinatorial pigeonhole/census arguments rooted in the nested structures.
• Switching and Contradiction: These forced inclusions allow the construction of a candidate rainbow Hamiltonian cycle. If the process fails, one is forced back into the known extremal configurations.
• Spectral Analysis: Stanley's inequality and spectral monotonicity under edge addition are essential. The identification of uniquely maximal spectral graphs (within specified families) is central, as is explicit calculation for key cases.

These techniques together guarantee tight thresholds, as well as full characterization of subcritical and critical families (Zhang et al., 31 Jan 2024).

5. Special Cases, Examples, and Connections

For small $n$, concrete examples clarify the correspondence between missing edges and rainbow cycle assignments:

• For $n=4$, all graphs with $e(G_i) \geq 5$ (i.e., complete or almost complete) admit rainbow Hamiltonian cycles; explicit coloring/descriptions suffice.
• For $n=5$, various deletions can occur, but systematic greedy labeling avoids color collision and produces a rainbow 5-cycle (Zhang et al., 31 Jan 2024).

Connections to generalizations:

• These results anchor the spectral and edge-count landscape for rainbow Hamiltonicity in multigraph systems, extending and sharpening the classical Dirac and Ore criteria in extremal graph theory.
• The symmetric structure of extremal graphs echoes the classic non-Hamiltonian threshold graphs, but here the rainbow constraint genuinely tightens possible exceptions.

6. Open Problems and Further Directions

Numerous avenues remain active, with central questions including:

• Rainbow Bondy-Size and Degree-Sum (Ore/Dirac) Extensions: Characterize rainbow Hamiltonicity when $e(G_i) \geq \binom{n-1}{2} + 1$ is allowed to reach equality (the "Bondy-size" regime), especially for $n \geq 6$: the only obstruction is conjectured to be the extremal threshold graph family (Zhang et al., 31 Jan 2024).
• Spectral and Degree Refinements: Seek rainbow versions of Dirac or Ore theorems in terms of minimum degree ($\delta$) or degree sum ($\sigma_2$); and consider analogues in terms of spectral radii for pancyclicity, matchings, and other spanning structures.
• Hypergraph and Multisystem Extensions: Analyze rainbow matching or Hamiltonicity in $k$-uniform hypergraph systems by spectral and codegree tools.
• Algorithmic and Structural Questions: Determine efficient algorithms for constructing rainbow Hamiltonian cycles under near-threshold conditions and characterize local vs. global obstacles.

Future research should deepen the extremal and probabilistic understanding of rainbow spanning structures, across graph systems and higher-order analogues.

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References:

• “Rainbow Hamiltonicity and the spectral radius” (Zhang et al., 31 Jan 2024)