Strictly Unfriendly $k$-Partitions: Sharp Degree Thresholds and ETH-Based Lower Bounds

2026-10-03

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Abstract

We present a complete complexity classification and fine-grained analysis for the Strictly Unfriendly $k$-Partition problem ($\text{SU}k\text{P}$), which asks whether the vertices of a graph can be partitioned into $k$ classes such that every vertex has strictly more neighbors in each of the other $k-1$ classes than in its own. We first establish a sharp tractability-intractability threshold with respect to the maximum degree $\Delta$: for $k \in \{2, 3\}$, $\text{SU}k\text{P}$ is solvable in polynomial time when $\Delta \le 2$, but becomes $\mathbf{NP}$-hard and ETH-hard immediately on subcubic graphs ($\Delta = 3$), resolving the degree limitations in prior work and establishing subcubic graphs as the precise frontier of intractability. Furthermore, under the Exponential Time Hypothesis (ETH), we establish the first fine-grained lower bounds via direct reductions from $(3,3)$-SAT. On general graphs, we establish a uniform lower bound across all partition parameters $k \ge 2$, revealing a striking complexity convergence where the core exponential complexity remains invariant despite technical divergences in gadget constructions. On subcubic graphs, we formally quantify the"cost of sparsity,"deriving explicit lower bound constants to demonstrate how enforced structural degree restrictions degrade reduction efficiency.

Problem

• Strictly Unfriendly k-Partition
• Complexity classification
• Fine-grained complexity
• Degree threshold
• Exponential Time Hypothesis

Innovation

• Strictly Unfriendly k-Partition
• Sharp Degree Thresholds
• Exponential Time Hypothesis
• Fine-grained Lower Bounds
• Complexity Classification