Aitao Zhang | Searcharxiv

arXiv2026

Disjoint unions and fixed-parameter algorithms for chromatic villainy

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.

math.CO↗

arXiv2026

Exact and extremal values of chromatic villainy

Chromatic villainy measures the largest number of vertices that must change color to repair a rearrangement of an optimal coloring while preserving every color multiplicity. Weak villainy allows different multiplicities in the repair. On the odd cycle $C_{2r+1}$, we determine the largest repair distance from arbitrary color assignments with a prescribed color-count vector (profile) $\mathbf m$ to proper colorings with any realizable target profile $\mathbf p$: it equals $2r+1-\sum_X\min\{p_X,\lceil m_X/2\rceil\}$. In particular, every feasible profile $(a,b,c)$ has ordinary and weak repair radius $\lfloor a/2\rfloor+\lfloor b/2\rfloor+\lfloor c/2\rfloor$, and $B(C_{2r+1})=r$ for $r\ge2$, answering a question of Clark et al. For arbitrary complete multipartite graphs, we give an arithmetic formula for ordinary villainy with at most $k^2$ candidates and an exact finite formula for weak villainy, where $k$ is the number of parts. Both values are fixed-parameter tractable in $k$ for binary-encoded part sizes. We also determine the maximum of each parameter among all $n$-vertex $k$-chromatic graphs. For $k\ge3$, the ordinary maximum is $\min\{n,4(n-k)\}$, except when $n=4(n-k)-1$ or $(n,k)\in\{(5,3),(7,3)\}$, when it is $n-1$. For $k\ge2$, the weak maximum is $n-\max\{\lceil n/k\rceil,2k-n\}-\varepsilon$, where $\varepsilon\in\{0,1\}$ is determined explicitly by $n$ and $k$. Both extrema are attained by complete multipartite graphs, and a weak extremizer needs at most three distinct part sizes.

math.CO↗