Disjoint unions and fixed-parameter algorithms for chromatic villainy

Source abstract

Chromatic villainy is the maximum, over rearrangements of optimal graph colorings, of the minimum Hamming distance to a proper coloring with the same color multiplicities. Weak villainy allows the repair to change these multiplicities. We give exact composition laws for disjoint unions and determine the eventual growth under repetition. Write $mG$ for the disjoint union of $m$ copies of $G$. For every graph $G$, with $k=χ(G)$, weak villainy satisfies $B_{\mathrm w}(mG)=mF_k(G)$ whenever $k\mid m$, where $F_k(G)$ is the maximum distance to a proper $k$-coloring over all assignments of $k$ color labels to vertices. Its bounded defect from $mF_k(G)$ is eventually periodic; we give an explicit stabilization bound and identify the least period, which divides $k$. For ordinary villainy, we obtain a computable rational slope $Λ(G)\ge F_k(G)$ such that $B(mG)=mΛ(G)+O_G(1)$, where the error is bounded independently of $m$ and eventually periodic, with a computable period and threshold. Consequently the difference between the two values is bounded exactly when $Λ(G)=F_k(G)$, and otherwise grows linearly. The slope is determined by a finite optimization using at most $k$ source assignments. We also answer the unrestricted superadditivity questions negatively: for every $C>0$, a connected graph can lose more than $C$ in weak villainy after union with a triangle, and some connected $3$-chromatic graph $G$ satisfies $B(G\sqcup G)<2B(G)-C$. Finally, both exact values on an $n$-vertex graph can be computed in time $f(B_{\mathrm w}(G))(n+1)^{O(1)}$, without either value being supplied. Threshold recognition is fixed-parameter tractable as well.