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Improved upper bounds on color reversal by local inversions

Kumud Singh Porte, RB Sandeep, Kamal Santra

TL;DR

This work improves fundamental bounds for color reversal under local inversions in bicolored graphs. By employing parity-aware decompositions and the Perfect Forest Theorem, the authors prove $cr(G) \le 4n-4$ when $n$ is even and $cr(G) \le 4n-3$ when $n$ is odd, and they bound the length needed to transform between any two colorings by $|w| \le \left\lfloor \frac{11n-3}{2}\right\rfloor$. They also establish $cr(S_n) \le 3n$ and $cr(K_n) \le 3n$ for star and complete graphs, respectively, via explicit constructions, all computable in polynomial time. The results hinge on partitioning graphs into induced odd trees through the Perfect Forest Theorem and on a suite of local-reversal lemmas, offering a cohesive framework for color-reversal and color-transformation in a broad class of graphs.

Abstract

We study the problem of color reversal in bicolored graphs under local inversions. A \emph{bicoloration} of a graph $G=(V,E)$ is a mapping $β: V \to \{-1,1\}$. A \emph{local inversion} at a vertex $v \in V$ consists of reversing the colors of all neighbors of $v$ and replacing the subgraph induced by these neighbors with its complement, while leaving $v$ and the rest of $G$ unchanged. Sabidussi (Discrete Mathematics, 1987) showed that any bicolored graph on $n$ vertices without isolated vertices can be color-reversed (that is, all vertex colors flipped while preserving the underlying graph) in at most $6n+3$ local inversions, and that any bicolored graph can be transformed into another bicolored graph on the same underlying graph in at most $9n$ local inversions. We improve both bounds: we prove that the first task can be accomplished in at most $4n-3$ local inversions, and the second in at most $ \left \lfloor \frac{11n-3}{2} \right \rfloor$ local inversions. Furthermore, we show that for stars and complete graphs, color reversal can be performed with at most $3n$ local inversions.

Improved upper bounds on color reversal by local inversions

TL;DR

This work improves fundamental bounds for color reversal under local inversions in bicolored graphs. By employing parity-aware decompositions and the Perfect Forest Theorem, the authors prove when is even and when is odd, and they bound the length needed to transform between any two colorings by . They also establish and for star and complete graphs, respectively, via explicit constructions, all computable in polynomial time. The results hinge on partitioning graphs into induced odd trees through the Perfect Forest Theorem and on a suite of local-reversal lemmas, offering a cohesive framework for color-reversal and color-transformation in a broad class of graphs.

Abstract

We study the problem of color reversal in bicolored graphs under local inversions. A \emph{bicoloration} of a graph is a mapping . A \emph{local inversion} at a vertex consists of reversing the colors of all neighbors of and replacing the subgraph induced by these neighbors with its complement, while leaving and the rest of unchanged. Sabidussi (Discrete Mathematics, 1987) showed that any bicolored graph on vertices without isolated vertices can be color-reversed (that is, all vertex colors flipped while preserving the underlying graph) in at most local inversions, and that any bicolored graph can be transformed into another bicolored graph on the same underlying graph in at most local inversions. We improve both bounds: we prove that the first task can be accomplished in at most local inversions, and the second in at most local inversions. Furthermore, we show that for stars and complete graphs, color reversal can be performed with at most local inversions.

Paper Structure

This paper contains 6 sections, 20 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Preliminaries
  4. General bounds
  5. Stars and complete graphs
  6. Concluding remarks

Key Result

proposition 1

For any bicolored graph $B = (G, \beta)$, $B_{aa}=B$, where $a$ is any vertex in $G$.

Figures (4)

  • Figure 1: Local complement of a graph $G$ with respect to vertex $a$.
  • Figure 2: Local complement of a graph with respect to the string $abca$.
  • Figure 3: Local inversion of a graph with respect to the string $abca$.
  • Figure 4: Partition of $T$ into $P_3s$ and $K_2$

Theorems & Definitions (30)

  • proposition 1: sabidussi1987color
  • proposition 2: sabidussi1987color
  • proposition 3: sabidussi1987color
  • proposition 4: sabidussi1987color
  • proposition 5: Perfect Forest Theorem caro2017twoscott2001induced
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 20 more