[Paper Review] Rainbow Pancyclicity in Graph Systems

Yangyang Cheng, Guanghui Wang+1 more|arXiv (Cornell University)|Sep 25, 2019

Graph theory and applications13 references4 citations

TL;DR

This paper proves an asymptotic version of Aharoni's conjecture on rainbow pancyclicity in graph systems: for sufficiently large n, any n graphs on n vertices, each with minimum degree at least (1/2 + ε)n, contain a rainbow Hamiltonian cycle. The proof uses a rainbow variant of the absorption technique, constructing a short rainbow cycle that can absorb any disjoint rainbow path with a new color, ultimately forming a rainbow Hamiltonian cycle.

ABSTRACT

Let $G_1,...,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $δ(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,...,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,...,G_n$ on the same vertex set of size $n$ with $δ(G_i)\ge (\frac{1}{2}+\varepsilon)n$, there exists a rainbow Hamiltonian cycle. Our main tool is the absorption technique. Additionally, we prove that with $δ(G_i)\geq \frac{n+1}{2}$ for each $i$, one can find rainbow cycles of length $3,...,n-1$.

Motivation & Objective

• To resolve an asymptotic version of Aharoni's conjecture on rainbow Hamiltonian cycles in graph systems.
• To establish conditions under which rainbow cycles of all lengths from 3 to n−1 exist in such systems.
• To demonstrate that the minimum degree threshold of n/2 in Aharoni's conjecture can be improved asymptotically to (1/2 + ε)n.
• To show that with δ(Gi) ≥ (n+1)/2, rainbow cycles of all lengths 3 to n−1 exist, and combined with later results, all lengths 3 to n.
• To develop and apply a rainbow absorption method for constructing rainbow Hamiltonian cycles in edge-colored multigraphs derived from graph systems.

Proposed method

• Construct a short rainbow cycle C using the absorption technique, where C can absorb any disjoint rainbow path P with a new color.
• Ensure that for any rainbow path P and unused color s, there exists an edge in C whose removal can be replaced by P via a colored path insertion.
• Use probabilistic methods and Chernoff bounds to show that a random family of 3-paths in a vertex set W can be pruned to a pairwise disjoint family W′ with sufficient size and low intersection.
• Apply concentration inequalities to guarantee that for each color s and vertex pair (x1,x2), there are many absorbing paths in C with color pattern (s, 3i−1, 3i−2, 3i).
• Use the union bound and Markov’s inequality to show that with positive probability, the required absorbing structure exists in C.
• Construct the final rainbow Hamiltonian cycle by first finding a rainbow Hamiltonian path in V(G) ∓ V(C), then absorbing it into C using the absorbing property.

Experimental results

Research questions

• RQ1 Can Aharoni’s conjecture on rainbow Hamiltonian cycles in graph systems be proven asymptotically?
• RQ2 What minimum degree condition ensures the existence of rainbow cycles of all lengths from 3 to n−1 in a system of n graphs?
• RQ3 Can the absorption technique be adapted to the rainbow setting to construct Hamiltonian cycles in edge-colored multigraphs?
• RQ4 Is the threshold δ(Gi) ≥ n/2 tight for rainbow Hamiltonian cycles, or can it be improved asymptotically?
• RQ5 What structural properties of the graph system allow for the existence of rainbow cycles of all lengths except possibly the Hamiltonian one?

Key findings

• For every ε > 0, there exists N such that for all n > N, any n graphs on n vertices with minimum degree at least (1/2 + ε)n contain a rainbow Hamiltonian cycle.
• With δ(Gi) ≥ (n+1)/2 for all i, the system contains rainbow cycles of all lengths from 3 to n−1.
• The minimum degree threshold of n/2 in Aharoni’s original conjecture is asymptotically improvable to (1/2 + ε)n.
• The absorbing structure used in the proof guarantees that any disjoint rainbow path with a new color can be inserted into the cycle.
• The construction ensures that the number of absorbing paths per color and vertex pair is at least Ω(εμ₁n), which is at least 1 for large n.
• The result, combined with Joos and Kim’s later proof of the full conjecture, confirms that systems with δ(Gi) ≥ (n+1)/2 contain rainbow cycles of all lengths 3 to n.