[Paper Review] Note on edge-colored graphs and digraphs without properly colored cycles

Gregory Gutin|ArXiv.org|Jul 31, 2007

Limits and Structures in Graph Theory5 references3 citations

TL;DR

This paper disproves a conjecture that every c-edge-colored undirected graph with minimum monochromatic degree d ≥ 1 must contain a properly colored cycle, by constructing recursive c-edge-colored graphs with arbitrarily high monochromatic degree and no properly colored cycles. The key contribution is establishing tight Θ(log n) bounds for the functions d(n,c) and d̲(n,c), showing they grow logarithmically with n for fixed c ≥ 2.

ABSTRACT

We study the following two functions: d(n,c) and $\vec{d}(n,c)$; d(n,c) ($\vec{d}(n,c)$) is the minimum number k such that every c-edge-colored undirected (directed) graph of order n and minimum monochromatic degree (out-degree) at least k has a properly colored cycle. Abouelaoualim et al. (2007) stated a conjecture which implies that d(n,c)=1. Using a recursive construction of c-edge-colored graphs with minimum monochromatic degree p and without properly colored cycles, we show that $d(n,c)\ge {1 \over c}(\log_cn -\log_c\log_cn)$ and, thus, the conjecture does not hold. In particular, this inequality significantly improves a lower bound on $\vec{d}(n,2)$ obtained by Gutin, Sudakov and Yeo in 1998.

Motivation & Objective

• To disprove a conjecture by Abouelaoualim et al. (2007) asserting that every c-edge-colored undirected graph with minimum monochromatic degree d ≥ 1 contains a properly colored cycle.
• To determine the exact asymptotic growth of d(n,c), the minimum k such that every c-edge-colored undirected graph of order n and minimum monochromatic degree ≥ k contains a properly colored cycle.
• To analyze the directed analogue d̲(n,c), defining the minimum k such that every c-edge-colored digraph of order n and minimum monochromatic out-degree ≥ k contains a properly colored cycle.
• To improve existing lower bounds on d̲(n,2), particularly surpassing the bound from Gutin, Sudakov, and Yeo (1998).
• To establish tight asymptotic Θ(log n) bounds for both d(n,c) and d̲(n,c) for every fixed c ≥ 2.

Proposed method

• Constructs recursive c-edge-colored graphs G(p₁, p₂, ..., p_c) where each vertex is connected via edges of color i to all vertices in a subgraph G with p_i − 1 for that color, starting from a single vertex.
• Uses induction on the sum s = p₁ + ... + p_c to prove that G(p₁, ..., p_c) has no properly colored cycles and has minimum monochromatic degree min{p_i}.
• Derives an upper bound on the order n(p₁, ..., p_c) ≤ s·2^s for s > 0, using recursive summation and induction.
• Applies the bound to the symmetric case p₁ = ... = p_c = p, showing that n_c(p) ≤ cp·c^{cp}, leading to a lower bound on p in terms of n.
• Establishes that d(n,c) ≥ (1/c)(log_c n − log_c log_c n) by showing existence of graphs with δ_mon ≥ d and no properly colored cycles.
• Uses a reduction from c-edge-colored digraphs to 2-edge-colored digraphs by merging colors to derive an upper bound on d̲(n,c) in terms of log₂n.

Experimental results

Research questions

• RQ1 Does every c-edge-colored undirected graph with minimum monochromatic degree d ≥ 1 necessarily contain a properly colored cycle?
• RQ2 What is the minimum k = d(n,c) such that every c-edge-colored undirected graph of order n and minimum monochromatic degree ≥ k contains a properly colored cycle?
• RQ3 What is the minimum k = d̲(n,c) such that every c-edge-colored directed graph of order n and minimum monochromatic out-degree ≥ k contains a properly colored cycle?
• RQ4 Can the lower bound on d̲(n,2) be improved beyond the existing bound of (1/4)log₂n + (1/8)log₂log₂n + Θ(1)?
• RQ5 Are d(n,c) and d̲(n,c) asymptotically Θ(log n) for every fixed c ≥ 2?

Key findings

• The conjecture by Abouelaoualim et al. (2007) is false, as there exist c-edge-colored graphs with arbitrarily high minimum monochromatic degree and no properly colored cycles.
• A recursive construction of c-edge-colored graphs G(p₁, ..., p_c) with no properly colored cycles and minimum monochromatic degree min{p_i} is established.
• The lower bound d(n,c) ≥ (1/c)(log_c n − log_c log_c n) is proven, significantly improving the previous lower bound for d̲(n,2).
• The bound d̲(n,2) ≥ (1/2)(log₂n − log₂log₂n) is derived, improving upon the (1/4)log₂n lower bound from Gutin, Sudakov, and Yeo (1998).
• It is shown that d(n,c) = Θ(log n) and d̲(n,c) = Θ(log n) for every fixed c ≥ 2, establishing tight asymptotic growth.
• An upper bound d̲(n,c) ≤ (1/⌊c/2⌋)(log₂n − (1/3)log₂log₂n + Θ(1)) is derived by reducing c-edge-colored digraphs to 2-edge-colored ones.