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A dichotomy theorem for $Γ$-switchable $H$-colouring on $m$-edge coloured graphs

Richard Brewster, Arnott Kidner, Gary MacGillivray

TL;DR

This work studies the complexity of Γ-switchable $H$-colouring on $m$-edge-coloured graphs, formalizing the Γ-switching operation and Γ-homomorphisms to a fixed target $H$. The authors introduce the switch graph $sw_{Γ}(H)$ to recast Γ-Hom-$H$ as a standard homomorphism problem, enabling a CSP framework. They establish a Hell–Nešetřil–style dichotomy for Abelian and transitive Γ: Γ-Hom-$H$ is polynomial-time solvable exactly when $H$ is Γ-homomorphically equivalent to a monochromatic bipartite graph (i.e., to $K_1$ or $K_2^i$); otherwise, the problem is NP-complete. For non-Abelian Γ, they reduce to an Abelianized instance via the commutator subgroup and blocks on colours, showing the general dichotomy follows from the Abelian case. The results illuminate the complexity landscape of homomorphism problems under a graph-switching symmetry and extend classical dichotomies to a coloured, switching-enabled setting, with potential implications for CSPs under colour-symmetry constraints.

Abstract

Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $Γ$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $π$ of $Γ$ permutes the colours of the edges incident with $x$ according to $π$. We investigate the complexity of whether there exists a sequence of switches that transforms a given $m$-edge coloured graph $G$ so that it has a colour-preserving homomorphism to a fixed $m$-edge coloured graph $H$ and give a dichotomy theorem in the case that $Γ$ acts transitively.

A dichotomy theorem for $Γ$-switchable $H$-colouring on $m$-edge coloured graphs

TL;DR

This work studies the complexity of Γ-switchable -colouring on -edge-coloured graphs, formalizing the Γ-switching operation and Γ-homomorphisms to a fixed target . The authors introduce the switch graph to recast Γ-Hom- as a standard homomorphism problem, enabling a CSP framework. They establish a Hell–Nešetřil–style dichotomy for Abelian and transitive Γ: Γ-Hom- is polynomial-time solvable exactly when is Γ-homomorphically equivalent to a monochromatic bipartite graph (i.e., to or ); otherwise, the problem is NP-complete. For non-Abelian Γ, they reduce to an Abelianized instance via the commutator subgroup and blocks on colours, showing the general dichotomy follows from the Abelian case. The results illuminate the complexity landscape of homomorphism problems under a graph-switching symmetry and extend classical dichotomies to a coloured, switching-enabled setting, with potential implications for CSPs under colour-symmetry constraints.

Abstract

Let be a graph in which each edge is assigned one of the colours , and let be a subgroup of . The operation of switching at a vertex of with respect to an element of permutes the colours of the edges incident with according to . We investigate the complexity of whether there exists a sequence of switches that transforms a given -edge coloured graph so that it has a colour-preserving homomorphism to a fixed -edge coloured graph and give a dichotomy theorem in the case that acts transitively.
Paper Structure (7 sections, 18 theorems, 8 equations, 2 figures)

This paper contains 7 sections, 18 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Switching homomorphisms and CSPs
  4. Main Results and Organization
  5. A dichotomy for Abelian $\Gamma$
  6. Group Abelianization
  7. Conclusion

Key Result

Theorem 1.1

Let $H$ be a fixed graph. Then the Hom-$H$ problem is polynomial-time solvable if $H$ is bipartite; otherwise, Hom-$H$ is NP-complete.

Figures (2)

  • Figure 1: An $(i,j)$-alternating cycle of length $12$ in $SH$ where $\pi$ has order $6$.
  • Figure 2: An $(i,j)$-alternating cycle of length $12(k-1)+2$ in $sw_{\Gamma}(H)$ where $\pi$ has order $6$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3: klo99bre09bre17macwar21
  • Theorem 1.4
  • Proposition 2.1: klo99bre09bre17kid21macwar21
  • Proposition 2.2
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5: bre_kid_mac21kid21
  • ...and 20 more