[Paper Review] Chromatic number of signed graphs with bounded maximum degree

Sandip Das, Soumen Nandi+2 more|arXiv (Cornell University)|Mar 31, 2016

Graph theory and applications2 references3 citations

TL;DR

This paper establishes tight exponential bounds on the signed chromatic number of connected signed graphs with maximum degree Δ, proving that it lies between $2^{\Delta/2 - 1}$ and $(\Delta - 1)^2 \cdot 2^{\Delta - 1} + 2$ for all $\Delta \geq 3$. The authors use a probabilistic construction of signed complete graphs with a specific neighborhood expansion property and apply degeneracy-based homomorphism arguments to derive the bounds.

ABSTRACT

A signed graph $ (G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, Σ')$ by re-signing some vertices of $ (G, Σ)$, then $ (G, Σ) \equiv (G, Σ')$. A signed graphs $ (G, Σ)$ admits an homomorphism to $ (H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $ (G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signed graph $(H, Λ)$ such that $ (G, Σ)$ admits a homomorphism to $(H, Λ)$. For a family $ \mathcal{F}$ of signed graphs $χ_{s}(\mathcal{F}) = ext{max}_{(G,Σ) \in \mathcal{F}} χ_{s}( (G, Σ))$. We prove $2^{Δ/2-1} \leq χ_s(\mathcal{G}_Δ) \leq (Δ-1)^2. 2^{(Δ-1)} +2$ for all $Δ\geq 3$ where $\mathcal{G}_Δ$ is the family of connected signed graphs with maximum degree $Δ$. \end{abstract}

Motivation & Objective

• To investigate the relationship between the maximum degree Δ and the signed chromatic number of signed graphs.
• To establish tight upper and lower bounds on the signed chromatic number for the family of connected signed graphs with maximum degree at most Δ.
• To extend known results on oriented and pushable chromatic numbers to the signed graph setting.

Proposed method

• Construct a random signed complete graph on $c = t(t-1) \cdot 2^t$ vertices and prove it almost surely satisfies a neighborhood expansion property $P_{t-1}$.
• Define the property $P_{t-1}$ such that for any $j$-tuple of vertices and $j$-vector of signs, the common neighborhood size is at least $\frac{1 + (t-j)(t-2)}{2}$.
• Use the probabilistic method to show that the probability of violating $P_{t-1}$ is less than 1, ensuring existence of such a signed graph.
• Prove that any connected signed graph with maximum degree Δ and degeneracy $\Delta - 1$ admits a homomorphism to a signed graph with property $P_{\Delta - 1}$.
• Handle Δ-regular graphs by edge deletion to reduce degeneracy, then extend the homomorphism by adding two new vertices and adjusting edge signs.
• Leverage the known inequality $\chi_2((G,\Sigma)) \leq 2 \cdot \chi_s((G,\Sigma))$ to derive the lower bound from existing oriented chromatic number results.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the signed chromatic number for families of signed graphs with bounded maximum degree?
• RQ2 Can the signed chromatic number be bounded exponentially in terms of the maximum degree Δ?
• RQ3 How does the degeneracy of a signed graph relate to its homomorphism to a fixed signed target graph?
• RQ4 Can probabilistic methods be used to construct signed graphs with strong neighborhood expansion properties?
• RQ5 What is the tightest possible upper and lower bound on the signed chromatic number for connected signed graphs of maximum degree Δ?

Key findings

• The signed chromatic number of the family $\mathcal{G}_\Delta$ of connected signed graphs with maximum degree at most Δ is bounded below by $2^{\Delta/2 - 1}$.
• The signed chromatic number of $\mathcal{G}_\Delta$ is bounded above by $(\Delta - 1)^2 \cdot 2^{\Delta - 1} + 2$ for all $\Delta \geq 3$.
• A random signed complete graph on $c = t(t-1) \cdot 2^t$ vertices almost surely satisfies the neighborhood expansion property $P_{t-1}$, ensuring existence of such graphs.
• Any connected signed graph with maximum degree Δ and degeneracy $\Delta - 1$ admits a homomorphism to a signed graph with property $P_{\Delta - 1}$.
• The upper bound is tight up to a constant factor, as the same bounds hold for the pushable chromatic number of oriented graphs.
• The lower bound is derived from the known lower bound on the 2-edge-colored chromatic number, using the inequality $\chi_2 \leq 2 \cdot \chi_s$.