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Inversion diameter and 2-edge-colored homomorphisms

Carmen Arana, Thomas Bellitto, Hector Buffière, Quentin Chuet, Théo Pierron, Amadeus Reinald

Abstract

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of inversions transforming one into the other. The inversion diameter diam(G) is the maximum such distance over all pairs of orientations of G. Through an equivalent formulation of inversions over 2-edge-colorings of G, we introduce the use of homomorphism-universal 2-edge-colored graphs to obtain bounds on the inversion diameter of various classes of graphs. Our first result upper bounds the inversion diameter by a linear function of the acyclic chromatic number, improving on the previous quadratic dependency. We then consider the inversion diameter of planar graphs, exhibiting a lower bound of 6, as well as new lower and upper bounds for those of a given girth, in particular settling the girth 7 case. We then show that any triangle-free graph G with maximum degree D satisfies diam(G) <= D + log D, making progress on the conjecture of Havet et al. that diam(G) <= D. Finally, we prove a general result about subdivisions: if a graph has inversion diameter k, any of its subdivisions has inversion diameter at most k + log k + 5.

Inversion diameter and 2-edge-colored homomorphisms

Abstract

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of inversions transforming one into the other. The inversion diameter diam(G) is the maximum such distance over all pairs of orientations of G. Through an equivalent formulation of inversions over 2-edge-colorings of G, we introduce the use of homomorphism-universal 2-edge-colored graphs to obtain bounds on the inversion diameter of various classes of graphs. Our first result upper bounds the inversion diameter by a linear function of the acyclic chromatic number, improving on the previous quadratic dependency. We then consider the inversion diameter of planar graphs, exhibiting a lower bound of 6, as well as new lower and upper bounds for those of a given girth, in particular settling the girth 7 case. We then show that any triangle-free graph G with maximum degree D satisfies diam(G) <= D + log D, making progress on the conjecture of Havet et al. that diam(G) <= D. Finally, we prove a general result about subdivisions: if a graph has inversion diameter k, any of its subdivisions has inversion diameter at most k + log k + 5.
Paper Structure (15 sections, 31 theorems, 1 equation, 10 figures, 1 table)

This paper contains 15 sections, 31 theorems, 1 equation, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions and techniques.
  3. Preliminaries
  4. Inversions of 2-edge-colored graphs
  5. Linear algebra expression of diam
  6. Bounding diam using homomorphisms
  7. Linear bound in the acyclic chromatic number
  8. Planar graphs
  9. General lower bound of 6
  10. Lower bound for girth 4
  11. Upper bound for girth 4
  12. Upper bounds for girth 5 and 6
  13. Upper bounds for girth 7
  14. Maximum degree bound for triangle-free graphs
  15. Subdivisions

Key Result

Theorem 1.1

The parameters $\mathop{\mathrm{diam}}\nolimits,\chi_a,\chi_s,\chi_o$ are functionally equivalent, and in particular: $\mathop{\mathrm{diam}}\nolimits \leq \frac{1}{3}(\chi_s^2+\chi_s+1), \frac{2}{3}(\chi_a^2+\chi_a+1), \chi_o^2-1$.

Figures (10)

  • Figure 1: A planar graph with a distinguished vertex $v$ used to construct gadget $G_3$
  • Figure 2: A planar graph with a distinguished vertex $v$ used to construct the gadget $G_4$
  • Figure 3: A planar graph that does not admit a solution with only vectors of weight $1$ or $3$
  • Figure 4: A girth 4 planar graph with inversion diameter 4, edges colored 1 are dotted.
  • Figure 5: Transformation of degree-3 vertices.
  • ...and 5 more figures

Theorems & Definitions (45)

  • Theorem 1.1: Theorem 1.2 in havet
  • Conjecture 1.2: Conjecture 6.4 in havet
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.4
  • Proposition 2.5
  • Definition 3.1
  • Theorem 3.2: raspaud1994good
  • Theorem 3.3
  • ...and 35 more